<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon" title="c1">X</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>let </b><font color="Maroon" title="c2">S</font> be   <a href="prob_1.html#NM3" title="PROB_1:NM.3">SigmaField</a> of <font color="Maroon" title="c1">X</font>;<br/><b>let </b><font color="Maroon" title="c3">M</font> be    <a href="measure1.html#M3" title="MEASURE1:mode.3">sigma_Measure</a> of <font color="Maroon" title="c2">S</font>;<br/><b>let </b><font color="Maroon" title="c4">g</font> be   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <font color="Maroon" title="c1">X</font>, <a href="numbers.html#K6" title="NUMBERS:func.6">ExtREAL</a> ;<br/><b>let </b><font color="Maroon" title="c5">F</font> be    <a href="seqfunc.html#M1" title="SEQFUNC:mode.1">Functional_Sequence</a> of <font color="Maroon" title="c1">X</font>, <a href="numbers.html#K6" title="NUMBERS:func.6">ExtREAL</a> ;<br/><b>let </b><font color="Maroon" title="c6">L</font> be   <a href="mesfunc5.html#NM1" title="MESFUNC5:NM.1">ExtREAL_sequence</a>;<br/>











<b>assume </b><b>that </b><br/><a NAME="E1:105"/><i><font color="Green" title="E99">A1</font></i>: 
<font color="Maroon" title="c4">g</font> <a href="mesfunc2.html#R1" title="MESFUNC2:pred.1">is_simple_func_in</a> <font color="Maroon" title="c2">S</font>
 <b>and </b><br/><a NAME="E2:105"/><i><font color="Green" title="E100">A2</font></i>: 
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">g</font> holds <br/>0 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">g</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Olive" title="b1">x</font>
 <b>and </b><br/><a NAME="E3:105"/><i><font color="Green" title="E101">A3</font></i>: 
for <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <font color="Maroon" title="c5">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Olive" title="b1">n</font> <a href="mesfunc2.html#R1" title="MESFUNC2:pred.1">is_simple_func_in</a> <font color="Maroon" title="c2">S</font>
 <b>and </b><br/><a NAME="E4:105"/><i><font color="Green" title="E102">A4</font></i>: 
for <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds   <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Olive" title="b1">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">g</font>
 <b>and </b><br/><a NAME="E5:105"/><i><font color="Green" title="E103">A5</font></i>: 
for <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <font color="Maroon" title="c5">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Olive" title="b1">n</font> is <a href="supinf_2.html#V6" title="SUPINF_2:attr.6">nonnegative</a>
 <b>and </b><br/><a NAME="E6:105"/><i><font color="Green" title="E104">A6</font></i>: 
for <font color="Olive" title="b1">n</font>, <font color="Olive" title="b2">m</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">m</font> holds <br/>for <font color="Olive" title="b3">x</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font>  st <font color="Olive" title="b3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">g</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Olive" title="b1">n</font></span>)</span> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Olive" title="b3">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Olive" title="b2">m</font></span>)</span> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Olive" title="b3">x</font>
 <b>and </b><br/><a NAME="E7:105"/><i><font color="Green" title="E105">A7</font></i>: 
for <font color="Olive" title="b1">x</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font>  st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">g</font> holds <br/>( <font color="Maroon" title="c5">F</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Olive" title="b1">x</font> is <a href="mesfunc5.html#V10" title="MESFUNC5:attr.10">convergent</a> &amp; <font color="Maroon" title="c4">g</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Olive" title="b1">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="mesfunc5.html#K2" title="MESFUNC5:func.2">lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Olive" title="b1">x</font></span>)</span> )
 <b>and </b><br/><a NAME="E8:105"/><i><font color="Green" title="E106">A8</font></i>: 
for <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <font color="Maroon" title="c6">L</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mesfunc5.html#K5" title="MESFUNC5:func.5">integral'</a> <font color="Maroon" title="c3">M</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Olive" title="b1">n</font></span>)</span>
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="105_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:105_1"/>
(  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  or  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">g</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="105_1_1"><b>suppose </b></a><a NAME="E1:105_1_1"/><i><font color="Green" title="E107">A9</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="105_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c7">n</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/><a NAME="E1:105_1_1_1"/>
(  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c7">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <font color="Maroon" title="c5">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c7">n</font> <a href="mesfunc2.html#R1" title="MESFUNC2:pred.1">is_simple_func_in</a> <font color="Maroon" title="c2">S</font> )
 <b>by </b><i><a class="txt" href="mesfunc5.html#E1:105_1_1"><i><font color="Green" title="E107">A9</font></i></a>, <a class="txt" href="mesfunc5.html#E4:105"><i><font color="Green" title="E102">A4</font></i></a>, <a class="txt" href="mesfunc5.html#E3:105"><i><font color="Green" title="E101">A3</font></i></a></i>;<br/><a NAME="E2:105_1_1_1"/><b>then </b>
 <a href="mesfunc5.html#K5" title="MESFUNC5:func.5">integral'</a> <font color="Maroon" title="c3">M</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c7">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 <b>by </b><i><a class="ref" href="mesfunc5.html#D14" target="_self" title="MESFUNC5:def.14">Def14</a></i>;<br/><b>hence </b><a NAME="E3:105_1_1_1"/>
<font color="Maroon" title="c6">L</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c7">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> 0
 <b>by </b><i><a class="txt" href="mesfunc5.html#E8:105"><i><font color="Green" title="E106">A8</font></i></a></i>;<br/></div><b>end;</b></div><a NAME="E3:105_1_1"/><b>then </b><i><font color="Green" title="E108">A10</font></i>: 
( <font color="Maroon" title="c6">L</font> is <a href="mesfunc5.html#V7" title="MESFUNC5:attr.7">convergent_to_finite_number</a> &amp;  <a href="mesfunc5.html#K2" title="MESFUNC5:func.2">lim</a> <font color="Maroon" title="c6">L</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> 0 )
 <b>by </b><i><a class="ref" href="mesfunc5.html#T58" target="_self" title="MESFUNC5:th.58">Th58</a></i>;<br/><b>hence </b><a NAME="E4:105_1_1"/>
<font color="Maroon" title="c6">L</font> is <a href="mesfunc5.html#V10" title="MESFUNC5:attr.10">convergent</a>
 <b>by </b><i><a class="ref" href="mesfunc5.html#D11" target="_self" title="MESFUNC5:def.11">Def11</a></i>;<br/><b>thus </b><a NAME="E5:105_1_1"/>
 <a href="mesfunc5.html#K5" title="MESFUNC5:func.5">integral'</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">g</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="mesfunc5.html#K2" title="MESFUNC5:func.2">lim</a> <font color="Maroon" title="c6">L</font>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E3:105_1_1"><i><font color="Green" title="E108">A10</font></i></a>, <a class="txt" href="mesfunc5.html#E1:105_1_1"><i><font color="Green" title="E107">A9</font></i></a>, <a class="ref" href="mesfunc5.html#D14" target="_self" title="MESFUNC5:def.14">Def14</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="105_1_2"><b>suppose </b></a><a NAME="E1:105_1_2"/><i><font color="Green" title="E107">A11</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">g</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ;<br/><div class="add"><a NAME="E2:105_1_2"/>
for <font color="Olive" title="b1">v</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">g</font> holds <br/>0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">g</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Olive" title="b1">v</font>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E2:105"><i><font color="Green" title="E100">A2</font></i></a></i>;<br/><b>then consider </b><font color="Maroon" title="c7">G</font> being   <a href="mesfunc2.html#NM1" title="MESFUNC2:NM.1">Finite_Sep_Sequence</a> of <font color="Maroon" title="c2">S</font>, <font color="Maroon" title="c8">a</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K6" title="NUMBERS:func.6">ExtREAL</a> <b> such that </b><br/><a NAME="E4:105_1_2"/><i><font color="Green" title="E108">A12</font></i>: 
( <font color="Maroon" title="c7">G</font>,<font color="Maroon" title="c8">a</font> <a href="mesfunc3.html#R1" title="MESFUNC3:pred.1">are_Re-presentation_of</a> <font color="Maroon" title="c4">g</font> &amp; <font color="Maroon" title="c8">a</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 &amp; ( for <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st 2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">n</font> &amp; <font color="Olive" title="b1">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c8">a</font> holds <br/>( 0 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c8">a</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Olive" title="b1">n</font> &amp; <font color="Maroon" title="c8">a</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>  ) ) )
 <b>by </b><i><a class="txt" href="mesfunc5.html#E1:105"><i><font color="Green" title="E99">A1</font></i></a>, <a class="ref" href="mesfunc3.html#T14" title="MESFUNC3:th.14">MESFUNC3:14</a></i>;<br/><a NAME="E5:105_1_2"/><i><font color="Green" title="E109">A13</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="105_1_2_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:105_1_2_1"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 ;<br/>

<a NAME="E2:105_1_2_1"/><b>then </b><i><font color="Green" title="E110">A14</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c8">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> 0
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E3:105_1_2_1"/><i><font color="Green" title="E111">A15</font></i>: 
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c7">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="105_1_2_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:105_1_2_1_1"/>
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c7">G</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ;<br/>

<b>then consider </b><font color="Maroon" title="c9">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:105_1_2_1_1"/><i><font color="Green" title="E112">A16</font></i>: 
<font color="Maroon" title="c9">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c7">G</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D1" title="XBOOLE_0:def.1">XBOOLE_0:def 1</a></i>;<br/>
<a NAME="E4:105_1_2_1_1"/>
 ex <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c7">G</font> &amp; <font color="Maroon" title="c9">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> )
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E3:105_1_2_1_1"><i><font color="Green" title="E112">A16</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<b>hence </b><a NAME="E5:105_1_2_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="mesfunc5.html#E2:105_1_2_1"><i><font color="Green" title="E110">A14</font></i></a>, <a class="txt" href="mesfunc5.html#E4:105_1_2"><i><font color="Green" title="E108">A12</font></i></a>, <a class="ref" href="mesfunc3.html#D1" title="MESFUNC3:def.1">MESFUNC3:def 1</a>, <a class="ref" href="finseq_1.html#T4" title="FINSEQ_1:th.4">FINSEQ_1:4</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:105_1_2_1"/>
 <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c7">G</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E1:105_1_2"><i><font color="Green" title="E107">A11</font></i></a>, <a class="ref" href="mesfunc3.html#D1" title="MESFUNC3:def.1">MESFUNC3:def 1</a>, <a class="txt" href="mesfunc5.html#E4:105_1_2"><i><font color="Green" title="E108">A12</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c9">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E6:105_1_2_1"/><i><font color="Green" title="E112">A17</font></i>: 
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c7">G</font></span>)</span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D1" title="XBOOLE_0:def.1">XBOOLE_0:def 1</a></i>;<br/>
<a NAME="E7:105_1_2_1"/>
 ex <font color="Olive" title="b1">Y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">Y</font> &amp; <font color="Olive" title="b1">Y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c7">G</font> )
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E6:105_1_2_1"><i><font color="Green" title="E112">A17</font></i></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/>
<b>hence </b><a NAME="E8:105_1_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="mesfunc5.html#E3:105_1_2_1"><i><font color="Green" title="E111">A15</font></i></a></i>;<br/>


</div><b>end;</b></div><b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b>$2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">a</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> $1;<br/><a NAME="E6:105_1_2"/><i><font color="Green" title="E110">A18</font></i>: 
for <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">a</font></span>)</span> holds <br/> ex <font color="Olive" title="b2">x</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="supinf_1.html#K3" title="SUPINF_1:func.3">REAL</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>,<font color="Olive" title="b2">x</font>]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="105_1_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c9">k</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:105_1_2_2"/>
<font color="Maroon" title="c9">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">a</font></span>)</span>
 ;<br/>

<a NAME="E2:105_1_2_2"/><b>then </b><i><font color="Green" title="E111">A19</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">k</font> &amp; <font color="Maroon" title="c9">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">a</font> &amp; <font color="Maroon" title="c9">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c8">a</font> )
 
<b>by </b><i><a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="105_1_2_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:105_1_2_2_1"/>
( <font color="Maroon" title="c9">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c9">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="105_1_2_2_1_1"><b>suppose </b></a><a NAME="E1:105_1_2_2_1_1"/><i><font color="Green" title="E112">A20</font></i>: 
<font color="Maroon" title="c9">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ;<br/><div class="add"><b>take </b><font color="Maroon" title="c10">x</font> = 0;<br/><b>thus </b><a NAME="E2:105_1_2_2_1_1"/>
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">a</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c9">k</font>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E1:105_1_2_2_1_1"><i><font color="Green" title="E112">A20</font></i></a>, <a class="txt" href="mesfunc5.html#E4:105_1_2"><i><font color="Green" title="E108">A12</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="105_1_2_2_1_2"><b>suppose </b></a><a NAME="E1:105_1_2_2_1_2"/>
<font color="Maroon" title="c9">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1
 ;<br/><div class="add"><a NAME="E2:105_1_2_2_1_2"/><b>then </b>
<font color="Maroon" title="c9">k</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1
 <b>by </b><i><a class="txt" href="mesfunc5.html#E2:105_1_2_2"><i><font color="Green" title="E111">A19</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/><a NAME="E3:105_1_2_2_1_2"/><b>then </b>
<font color="Maroon" title="c9">k</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E4:105_1_2_2_1_2"/><b>then </b>
( 0 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c8">a</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c9">k</font> &amp; <font color="Maroon" title="c8">a</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c9">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>  )
 <b>by </b><i><a class="txt" href="mesfunc5.html#E4:105_1_2"><i><font color="Green" title="E108">A12</font></i></a>, <a class="txt" href="mesfunc5.html#E2:105_1_2_2"><i><font color="Green" title="E111">A19</font></i></a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c10">x</font> = <font color="Maroon" title="c8">a</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c9">k</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="supinf_1.html#K3" title="SUPINF_1:func.3">REAL</a>  <b>by </b><i><a class="ref" href="xxreal_0.html#T14" title="XXREAL_0:th.14">XXREAL_0:14</a></i>;<br/><b>take </b>
<font color="Maroon" title="c10">x</font>
;<br/><b>thus </b><a NAME="E6:105_1_2_2_1_2"/>
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">a</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c9">k</font>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><b>consider </b><font color="Maroon" title="c9">a1</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="supinf_1.html#K3" title="SUPINF_1:func.3">REAL</a> <b> such that </b><br/><a NAME="E8:105_1_2"/><i><font color="Green" title="E111">A21</font></i>: 
(  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c9">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">a</font></span>)</span> &amp; ( for <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">a</font></span>)</span> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>,<font color="Maroon" title="c9">a1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">k</font>] ) )
 <b>from </b><i><a class="ref" href="finseq_1.html#S5" title="FINSEQ_1:sch.5">FINSEQ_1:sch 5</a>(<a class="txt" href="mesfunc5.html#E6:105_1_2"><i><font color="Green" title="E110">A18</font></i></a>);<br/></i><a NAME="E9:105_1_2"/><i><font color="Green" title="E112">A22</font></i>: 
2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">a</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="105_1_2_3"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:105_1_2_3"/>
not 2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">a</font>
 ;<br/>

<a NAME="E2:105_1_2_3"/><b>then </b>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E5:105_1_2"><i><font color="Green" title="E109">A13</font></i></a>, <a class="ref" href="nat_1.html#T23" title="NAT_1:th.23">NAT_1:23</a></i>;<br/>
<a NAME="E3:105_1_2_3"/><b>then </b>
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c8">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">1</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a>, <a class="ref" href="finseq_1.html#T4" title="FINSEQ_1:th.4">FINSEQ_1:4</a></i>;<br/>
<a NAME="E4:105_1_2_3"/><b>then </b><i><font color="Green" title="E113">A23</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c7">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">1</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E4:105_1_2"><i><font color="Green" title="E108">A12</font></i></a>, <a class="ref" href="mesfunc3.html#D1" title="MESFUNC3:def.1">MESFUNC3:def 1</a></i>;<br/>
<a NAME="E5:105_1_2_3"/><b>then </b><i><font color="Green" title="E114">A24</font></i>: 
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c7">G</font>
 
<b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<i><font color="Green" title="E115">A25</font></i>:  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">g</font> = 
 <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c7">G</font></span>)</span>
<b>by </b><i><a class="txt" href="mesfunc5.html#E4:105_1_2"><i><font color="Green" title="E108">A12</font></i></a>, <a class="ref" href="mesfunc3.html#D1" title="MESFUNC3:def.1">MESFUNC3:def 1</a></i>
<br/>.= 
 <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
<b>by </b><i><a class="txt" href="mesfunc5.html#E4:105_1_2_3"><i><font color="Green" title="E113">A23</font></i></a>, <a class="ref" href="funct_1.html#T14" title="FUNCT_1:th.14">FUNCT_1:14</a></i>
<br/>.= 
<font color="Maroon" title="c7">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1
<b>by </b><i><a class="ref" href="zfmisc_1.html#T31" title="ZFMISC_1:th.31">ZFMISC_1:31</a></i>
;<br/>
<b>then consider </b><font color="Maroon" title="c10">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E8:105_1_2_3"/><i><font color="Green" title="E116">A26</font></i>: 
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1
 <b>by </b><i><a class="txt" href="mesfunc5.html#E1:105_1_2"><i><font color="Green" title="E107">A11</font></i></a>, <a class="ref" href="xboole_0.html#D1" title="XBOOLE_0:def.1">XBOOLE_0:def 1</a></i>;<br/>
<a NAME="E9:105_1_2_3"/>
<font color="Maroon" title="c4">g</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c10">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E4:105_1_2"><i><font color="Green" title="E108">A12</font></i></a>, <a class="txt" href="mesfunc5.html#E8:105_1_2_3"><i><font color="Green" title="E116">A26</font></i></a>, <a class="txt" href="mesfunc5.html#E5:105_1_2_3"><i><font color="Green" title="E114">A24</font></i></a>, <a class="ref" href="mesfunc3.html#D1" title="MESFUNC3:def.1">MESFUNC3:def 1</a></i>;<br/>
<b>hence </b><a NAME="E10:105_1_2_3"/>
contradiction
 <b>by </b><i><a class="txt" href="mesfunc5.html#E2:105"><i><font color="Green" title="E100">A2</font></i></a>, <a class="txt" href="mesfunc5.html#E8:105_1_2_3"><i><font color="Green" title="E116">A26</font></i></a>, <a class="txt" href="mesfunc5.html#E6:105_1_2_3"><i><font color="Green" title="E115">A25</font></i></a></i>;<br/>


</div><b>end;</b></div><a NAME="E10:105_1_2"/><b>then </b><i><font color="Green" title="E113">A27</font></i>: 
2 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c9">a1</font>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E8:105_1_2"><i><font color="Green" title="E111">A21</font></i></a></i>;<br/><a NAME="E11:105_1_2"/><b>then </b>
( <font color="Maroon" title="c9">a1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">a</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> 2 &amp; 2 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c8">a</font> )
 <b>by </b><i><a class="txt" href="mesfunc5.html#E8:105_1_2"><i><font color="Green" title="E111">A21</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E12:105_1_2"/><b>then </b><i><font color="Green" title="E114">A28</font></i>: 
<font color="Maroon" title="c9">a1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 2 <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c8">a</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> 1
 <b>by </b><i><a class="txt" href="mesfunc5.html#E4:105_1_2"><i><font color="Green" title="E108">A12</font></i></a></i>;<br/><a NAME="E13:105_1_2"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">a</font>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E9:105_1_2"><i><font color="Green" title="E112">A22</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E14:105_1_2"/><b>then </b>
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">a</font></span>)</span>
 ;<br/><a NAME="E15:105_1_2"/><b>then </b>
<font color="Maroon" title="c8">a</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">a1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1
 <b>by </b><i><a class="txt" href="mesfunc5.html#E8:105_1_2"><i><font color="Green" title="E111">A21</font></i></a></i>;<br/><a NAME="E16:105_1_2"/><b>then </b>
( not <font color="Maroon" title="c9">a1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 2 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">a1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c9">a1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 2 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c9">a1</font> )
 <b>by </b><i><a class="txt" href="mesfunc5.html#E12:105_1_2"><i><font color="Green" title="E114">A28</font></i></a>, <a class="txt" href="mesfunc5.html#E10:105_1_2"><i><font color="Green" title="E113">A27</font></i></a>, <a class="ref" href="funct_1.html#T12" title="FUNCT_1:th.12">FUNCT_1:12</a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c10">RINGA</font> = <span class="p1">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c9">a1</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">a1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> <a href="membered.html#V3" title="MEMBERED:attr.3">real-membered</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  <b>by </b><i><a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/><b>set </b><font color="Maroon" title="c11">alpha</font> =  <a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <span class="p1">(<span class="default"><a href="seq_4.html#K3" title="SEQ_4:func.3">min</a> <font color="Maroon" title="c10">RINGA</font></span>)</span>;<br/><b>set </b><font color="Maroon" title="c12">beta</font> =  <a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <span class="p1">(<span class="default"><a href="pre_circ.html#K1" title="PRE_CIRC:func.1">max</a> <font color="Maroon" title="c10">RINGA</font></span>)</span>;<br/><b>reconsider </b><font color="Maroon" title="c13">beta1</font> =  <a href="pre_circ.html#K1" title="PRE_CIRC:func.1">max</a> <font color="Maroon" title="c10">RINGA</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> <b>by </b><i><a class="ref" href="xreal_0.html#D1" title="XREAL_0:def.1">XREAL_0:def 1</a></i>;<br/><a NAME="E19:105_1_2"/>
 <a href="seq_4.html#K3" title="SEQ_4:func.3">min</a> <font color="Maroon" title="c10">RINGA</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">RINGA</font>
 <b>by </b><i><a class="ref" href="pre_circ.html#D4" title="PRE_CIRC:def.4">PRE_CIRC:def 4</a></i>;<br/><a NAME="E20:105_1_2"/><b>then </b><i><font color="Green" title="E115">A29</font></i>: 
(  <a href="seq_4.html#K3" title="SEQ_4:func.3">min</a> <font color="Maroon" title="c10">RINGA</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c9">a1</font> &amp; not  <a href="seq_4.html#K3" title="SEQ_4:func.3">min</a> <font color="Maroon" title="c10">RINGA</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">a1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 <b>by </b><i><a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/><b>then consider </b><font color="Maroon" title="c14">i</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E22:105_1_2"/><i><font color="Green" title="E116">A30</font></i>: 
( <font color="Maroon" title="c14">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c9">a1</font> &amp;  <a href="seq_4.html#K3" title="SEQ_4:func.3">min</a> <font color="Maroon" title="c10">RINGA</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">a1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">i</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c15">i</font> = <font color="Maroon" title="c14">i</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="txt" href="mesfunc5.html#E22:105_1_2"><i><font color="Green" title="E116">A30</font></i></a></i>;<br/><a NAME="E24:105_1_2"/><i><font color="Green" title="E117">A31</font></i>: 
<font color="Maroon" title="c15">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1
 <b>by </b><i><a class="txt" href="mesfunc5.html#E20:105_1_2"><i><font color="Green" title="E115">A29</font></i></a>, <a class="txt" href="mesfunc5.html#E22:105_1_2"><i><font color="Green" title="E116">A30</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><a NAME="E25:105_1_2"/>
<font color="Maroon" title="c15">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">a1</font></span>)</span>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E22:105_1_2"><i><font color="Green" title="E116">A30</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E26:105_1_2"/><b>then </b>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c15">i</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/><a NAME="E27:105_1_2"/><b>then </b>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c15">i</font>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E24:105_1_2"><i><font color="Green" title="E117">A31</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/><a NAME="E28:105_1_2"/><b>then </b><i><font color="Green" title="E118">A32</font></i>: 
1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c15">i</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E29:105_1_2"/><i><font color="Green" title="E119">A33</font></i>: 
<font color="Maroon" title="c4">g</font> is <a href="seq_1.html#V1" title="SEQ_1:attr.1">finite</a>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E1:105"><i><font color="Green" title="E99">A1</font></i></a>, <a class="ref" href="mesfunc2.html#D5" title="MESFUNC2:def.5">MESFUNC2:def 5</a></i>;<br/><a NAME="E30:105_1_2"/><i><font color="Green" title="E120">A34</font></i>: 
for <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds   <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="mesfunc1.html#K6" title="MESFUNC1:func.6">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Olive" title="b1">n</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">g</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="105_1_2_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c16">n</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/>

<a NAME="E1:105_1_2_4"/><i><font color="Green" title="E121">A35</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c16">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">g</font>
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E4:105"><i><font color="Green" title="E102">A4</font></i></a></i>;<br/>
<a NAME="E2:105_1_2_4"/>
( <font color="Maroon" title="c4">g</font> is <a href="mesfunc5.html#V4" title="MESFUNC5:attr.4">without+infty</a> &amp; <font color="Maroon" title="c4">g</font> is <a href="mesfunc5.html#V3" title="MESFUNC5:attr.3">without-infty</a> )
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E1:105"><i><font color="Green" title="E99">A1</font></i></a>, <a class="ref" href="mesfunc5.html#T20" target="_self" title="MESFUNC5:th.20">Th20</a></i>;<br/>
<a NAME="E3:105_1_2_4"/><b>then </b>
( not  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c4">g</font> &amp; not  <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c4">g</font> )
 
<b>by </b><i><a class="ref" href="mesfunc5.html#D3" target="_self" title="MESFUNC5:def.3">Def3</a>, <a class="ref" href="mesfunc5.html#D4" target="_self" title="MESFUNC5:def.4">Def4</a></i>;<br/>
<a NAME="E4:105_1_2_4"/><b>then </b><i><font color="Green" title="E122">A36</font></i>: 
( <font color="Maroon" title="c4">g</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <font color="Maroon" title="c4">g</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 
<b>by </b><i><a class="ref" href="funct_1.html#T142" title="FUNCT_1:th.142">FUNCT_1:142</a></i>;<br/>
<a NAME="E5:105_1_2_4"/>
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="mesfunc1.html#K6" title="MESFUNC1:func.6">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c16">n</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c16">n</font></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">g</font></span>)</span></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c16">n</font></span>)</span> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p4"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p4"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c16">n</font></span>)</span> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p4"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p4"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="mesfunc1.html#D4" title="MESFUNC1:def.4">MESFUNC1:def 4</a></i>;<br/>
<b>hence </b><a NAME="E6:105_1_2_4"/>
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="mesfunc1.html#K6" title="MESFUNC1:func.6">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c16">n</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">g</font>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E1:105_1_2_4"><i><font color="Green" title="E121">A35</font></i></a>, <a class="txt" href="mesfunc5.html#E4:105_1_2_4"><i><font color="Green" title="E122">A36</font></i></a></i>;<br/>


</div><b>end;</b></div><a NAME="E31:105_1_2"/><i><font color="Green" title="E121">A37</font></i>: 
for <font color="Olive" title="b1">e</font> being  <a href="supinf_1.html#NM3" title="SUPINF_1:NM.3">R_eal</a>  st  <a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> 0 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">e</font> &amp; <font color="Olive" title="b1">e</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <span class="p1">(<span class="default"><a href="seq_4.html#K3" title="SEQ_4:func.3">min</a> <font color="Maroon" title="c10">RINGA</font></span>)</span> holds <br/> ex <font color="Olive" title="b2">H</font> being  <a href="prob_1.html#NM2" title="PROB_1:NM.2">SetSequence</a> of <font color="Maroon" title="c1">X</font> ex <font color="Olive" title="b3">MF</font> being  <a href="mesfunc5.html#NM1" title="MESFUNC5:NM.1">ExtREAL_sequence</a> st <br/>( ( for <font color="Olive" title="b4">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <font color="Olive" title="b2">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b4">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mesfunc1.html#K15" title="MESFUNC1:func.15">less_dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="mesfunc1.html#K6" title="MESFUNC1:func.6">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Olive" title="b4">n</font></span>)</span></span>)</span>,<font color="Olive" title="b1">e</font> ) &amp; ( for <font color="Olive" title="b4">n</font>, <font color="Olive" title="b5">m</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b4">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b5">m</font> holds <br/><font color="Olive" title="b2">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b4">n</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b2">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b5">m</font> ) &amp; ( for <font color="Olive" title="b4">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <font color="Olive" title="b2">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b4">n</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">g</font> ) &amp; ( for <font color="Olive" title="b4">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <font color="Olive" title="b3">MF</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Olive" title="b4">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">M</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b4">n</font></span>)</span> ) &amp; <font color="Maroon" title="c3">M</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">g</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_2.html#K8" title="SUPINF_2:func.8">sup</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K19" title="SUPINF_2:func.19">rng</a> <font color="Olive" title="b3">MF</font></span>)</span> &amp; ( for <font color="Olive" title="b4">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <font color="Olive" title="b2">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b4">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">S</font> ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="105_1_2_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c16">e</font> be   <a href="supinf_1.html#NM3" title="SUPINF_1:NM.3">R_eal</a>;<br/>

<b>assume </b><a NAME="E1:105_1_2_5"/><i><font color="Green" title="E122">A38</font></i>: 
(  <a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> 0 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c16">e</font> &amp; <font color="Maroon" title="c16">e</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <span class="p1">(<span class="default"><a href="seq_4.html#K3" title="SEQ_4:func.3">min</a> <font color="Maroon" title="c10">RINGA</font></span>)</span> )
 ;<br/>

<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>) <b>-&gt; </b>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <font color="Maroon" title="c1">X</font> =  <a href="mesfunc1.html#K15" title="MESFUNC1:func.15">less_dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="mesfunc1.html#K6" title="MESFUNC1:func.6">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> $1</span>)</span></span>)</span>,<font color="Maroon" title="c16">e</font>;<br/>
<b>consider </b><font color="Maroon" title="c17">H</font> being   <a href="prob_1.html#NM2" title="PROB_1:NM.2">SetSequence</a> of <font color="Maroon" title="c1">X</font><b> such that </b><br/><a NAME="E3:105_1_2_5"/><i><font color="Green" title="E123">A39</font></i>: 
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon" title="c17">H</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">n</font>)
 <b>from </b><i><a class="ref" href="kurato_2.html#S1" title="KURATO_2:sch.1">KURATO_2:sch 1</a>();<br/></i>
<div><i><font color="Green" title="E124">A40</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="105_1_2_5_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c18">n</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/><a NAME="E1:105_1_2_5_1"/>
<font color="Maroon" title="c18">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 <b>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/><b>hence </b><a NAME="E2:105_1_2_5_1"/>
<font color="Maroon" title="c17">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c18">n</font>)
 <b>by </b><i><a class="txt" href="mesfunc5.html#E3:105_1_2_5"><i><font color="Green" title="E123">A39</font></i></a></i>;<br/></div><b>end;</b></div>
<a NAME="E5:105_1_2_5"/><i><font color="Green" title="E125">A41</font></i>: 
for <font color="Olive" title="b1">n</font>, <font color="Olive" title="b2">m</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">m</font> holds <br/><font color="Maroon" title="c17">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c17">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">m</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="105_1_2_5_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c18">n</font>, <font color="Maroon" title="c19">m</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:105_1_2_5_2"/><i><font color="Green" title="E126">A42</font></i>: 
<font color="Maroon" title="c18">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c19">m</font>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="105_1_2_5_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c20">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:105_1_2_5_2_1"/>
<font color="Maroon" title="c20">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c17">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">n</font>
 ;<br/><a NAME="E2:105_1_2_5_2_1"/><b>then </b><i><font color="Green" title="E127">A43</font></i>: 
<font color="Maroon" title="c20">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="mesfunc1.html#K15" title="MESFUNC1:func.15">less_dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="mesfunc1.html#K6" title="MESFUNC1:func.6">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c18">n</font></span>)</span></span>)</span>,<font color="Maroon" title="c16">e</font>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E4:105_1_2_5"><i><font color="Green" title="E124">A40</font></i></a></i>;<br/><a NAME="E3:105_1_2_5_2_1"/><b>then </b><i><font color="Green" title="E128">A44</font></i>: 
( <font color="Maroon" title="c20">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="mesfunc1.html#K6" title="MESFUNC1:func.6">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c18">n</font></span>)</span></span>)</span> &amp;  ex <font color="Olive" title="b1">y</font> being  <a href="supinf_1.html#NM3" title="SUPINF_1:NM.3">R_eal</a> st <br/>( <font color="Olive" title="b1">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="mesfunc1.html#K6" title="MESFUNC1:func.6">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c18">n</font></span>)</span></span>)</span> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c20">x</font> &amp; <font color="Olive" title="b1">y</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c16">e</font> ) )
 <b>by </b><i><a class="ref" href="mesfunc1.html#D12" title="MESFUNC1:def.12">MESFUNC1:def 12</a></i>;<br/><a NAME="E4:105_1_2_5_2_1"/><b>then </b><i><font color="Green" title="E129">A45</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="mesfunc1.html#K6" title="MESFUNC1:func.6">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c18">n</font></span>)</span></span>)</span> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c20">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c20">x</font></span>)</span> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c18">n</font></span>)</span> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c20">x</font></span>)</span>
 <b>by </b><i><a class="ref" href="mesfunc1.html#D4" title="MESFUNC1:def.4">MESFUNC1:def 4</a></i>;<br/><a NAME="E5:105_1_2_5_2_1"/><i><font color="Green" title="E130">A46</font></i>: 
 ex <font color="Olive" title="b1">y1</font> being  <a href="supinf_1.html#NM3" title="SUPINF_1:NM.3">R_eal</a> st <br/>( <font color="Olive" title="b1">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="mesfunc1.html#K6" title="MESFUNC1:func.6">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c18">n</font></span>)</span></span>)</span> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c20">x</font> &amp; <font color="Olive" title="b1">y1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c16">e</font> )
 <b>by </b><i><a class="txt" href="mesfunc5.html#E2:105_1_2_5_2_1"><i><font color="Green" title="E127">A43</font></i></a>, <a class="ref" href="mesfunc1.html#D12" title="MESFUNC1:def.12">MESFUNC1:def 12</a></i>;<br/><a NAME="E6:105_1_2_5_2_1"/><i><font color="Green" title="E131">A47</font></i>: 
(  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="mesfunc1.html#K6" title="MESFUNC1:func.6">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c18">n</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">g</font> &amp;  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="mesfunc1.html#K6" title="MESFUNC1:func.6">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c19">m</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">g</font> )
 <b>by </b><i><a class="txt" href="mesfunc5.html#E30:105_1_2"><i><font color="Green" title="E120">A34</font></i></a></i>;<br/><a NAME="E7:105_1_2_5_2_1"/><b>then </b><i><font color="Green" title="E132">A48</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="mesfunc1.html#K6" title="MESFUNC1:func.6">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c19">m</font></span>)</span></span>)</span> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c20">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c20">x</font></span>)</span> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c19">m</font></span>)</span> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c20">x</font></span>)</span>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E3:105_1_2_5_2_1"><i><font color="Green" title="E128">A44</font></i></a>, <a class="ref" href="mesfunc1.html#D4" title="MESFUNC1:def.4">MESFUNC1:def 4</a></i>;<br/><a NAME="E8:105_1_2_5_2_1"/><i><font color="Green" title="E133">A49</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c18">n</font></span>)</span> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c20">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c19">m</font></span>)</span> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c20">x</font>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E1:105_1_2_5_2"><i><font color="Green" title="E126">A42</font></i></a>, <a class="txt" href="mesfunc5.html#E3:105_1_2_5_2_1"><i><font color="Green" title="E128">A44</font></i></a>, <a class="txt" href="mesfunc5.html#E6:105_1_2_5_2_1"><i><font color="Green" title="E131">A47</font></i></a>, <a class="txt" href="mesfunc5.html#E6:105"><i><font color="Green" title="E104">A6</font></i></a></i>;<br/><a NAME="E9:105_1_2_5_2_1"/>
<span class="p1"><a href="extreal1.html#K4" title="EXTREAL1:func.4">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c20">x</font></span>)</span></span><a href="extreal1.html#K4" title="EXTREAL1:func.4">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a> 
 <b>by </b><i><a class="txt" href="mesfunc5.html#E6:105_1_2_5_2_1"><i><font color="Green" title="E131">A47</font></i></a>, <a class="txt" href="mesfunc5.html#E3:105_1_2_5_2_1"><i><font color="Green" title="E128">A44</font></i></a>, <a class="txt" href="mesfunc5.html#E29:105_1_2"><i><font color="Green" title="E119">A33</font></i></a>, <a class="ref" href="mesfunc2.html#D1" title="MESFUNC2:def.1">MESFUNC2:def 1</a></i>;<br/><a NAME="E10:105_1_2_5_2_1"/><b>then </b>
( <font color="Maroon" title="c4">g</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c20">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a>  &amp; <font color="Maroon" title="c4">g</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c20">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>  )
 <b>by </b><i><a class="ref" href="extreal2.html#T67" title="EXTREAL2:th.67">EXTREAL2:67</a></i>;<br/><a NAME="E11:105_1_2_5_2_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="mesfunc1.html#K6" title="MESFUNC1:func.6">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c19">m</font></span>)</span></span>)</span> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c20">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="mesfunc1.html#K6" title="MESFUNC1:func.6">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c18">n</font></span>)</span></span>)</span> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c20">x</font>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E4:105_1_2_5_2_1"><i><font color="Green" title="E129">A45</font></i></a>, <a class="txt" href="mesfunc5.html#E7:105_1_2_5_2_1"><i><font color="Green" title="E132">A48</font></i></a>, <a class="txt" href="mesfunc5.html#E8:105_1_2_5_2_1"><i><font color="Green" title="E133">A49</font></i></a>, <a class="ref" href="supinf_2.html#T15" title="SUPINF_2:th.15">SUPINF_2:15</a></i>;<br/><a NAME="E12:105_1_2_5_2_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="mesfunc1.html#K6" title="MESFUNC1:func.6">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c19">m</font></span>)</span></span>)</span> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c20">x</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c16">e</font>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E5:105_1_2_5_2_1"><i><font color="Green" title="E130">A46</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E13:105_1_2_5_2_1"/><b>then </b>
<font color="Maroon" title="c20">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="mesfunc1.html#K15" title="MESFUNC1:func.15">less_dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="mesfunc1.html#K6" title="MESFUNC1:func.6">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c19">m</font></span>)</span></span>)</span>,<font color="Maroon" title="c16">e</font>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E3:105_1_2_5_2_1"><i><font color="Green" title="E128">A44</font></i></a>, <a class="txt" href="mesfunc5.html#E6:105_1_2_5_2_1"><i><font color="Green" title="E131">A47</font></i></a>, <a class="ref" href="mesfunc1.html#D12" title="MESFUNC1:def.12">MESFUNC1:def 12</a></i>;<br/><b>hence </b><a NAME="E14:105_1_2_5_2_1"/>
<font color="Maroon" title="c20">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c17">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">m</font>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E4:105_1_2_5"><i><font color="Green" title="E124">A40</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:105_1_2_5_2"/>
<font color="Maroon" title="c17">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">n</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c17">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">m</font>
 <b>by </b><i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/>


</div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="105_1_2_5_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c18">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:105_1_2_5_3"/>
<font color="Maroon" title="c18">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="kurato_2.html#K4" title="KURATO_2:func.4">lim_inf</a> <font color="Maroon" title="c17">H</font>
 ;<br/><a NAME="E2:105_1_2_5_3"/><b>then </b>
<font color="Maroon" title="c18">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="prob_1.html#K2" title="PROB_1:func.2">Union</a> <span class="p1">(<span class="default"><a href="setlim_1.html#K1" title="SETLIM_1:func.1">inferior_setsequence</a> <font color="Maroon" title="c17">H</font></span>)</span>
 <b>by </b><i><a class="ref" href="setlim_1.html#D4" title="SETLIM_1:def.4">SETLIM_1:def 4</a></i>;<br/><b>then consider </b><font color="Maroon" title="c19">V</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E4:105_1_2_5_3"/><i><font color="Green" title="E126">A50</font></i>: 
( <font color="Maroon" title="c18">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c19">V</font> &amp; <font color="Maroon" title="c19">V</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><a href="setlim_1.html#K1" title="SETLIM_1:func.1">inferior_setsequence</a> <font color="Maroon" title="c17">H</font></span>)</span> )
 <b>by </b><i><a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/><b>consider </b><font color="Maroon" title="c20">n</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E6:105_1_2_5_3"/><i><font color="Green" title="E127">A51</font></i>: 
( <font color="Maroon" title="c20">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="setlim_1.html#K1" title="SETLIM_1:func.1">inferior_setsequence</a> <font color="Maroon" title="c17">H</font></span>)</span> &amp; <font color="Maroon" title="c19">V</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="setlim_1.html#K1" title="SETLIM_1:func.1">inferior_setsequence</a> <font color="Maroon" title="c17">H</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">n</font> )
 <b>by </b><i><a class="txt" href="mesfunc5.html#E4:105_1_2_5_3"><i><font color="Green" title="E126">A50</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c21">n</font> = <font color="Maroon" title="c20">n</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="txt" href="mesfunc5.html#E6:105_1_2_5_3"><i><font color="Green" title="E127">A51</font></i></a></i>;<br/><a NAME="E8:105_1_2_5_3"/>
<font color="Maroon" title="c18">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c17">H</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 0</span>)</span>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E4:105_1_2_5_3"><i><font color="Green" title="E126">A50</font></i></a>, <a class="txt" href="mesfunc5.html#E6:105_1_2_5_3"><i><font color="Green" title="E127">A51</font></i></a>, <a class="ref" href="setlim_1.html#T24" title="SETLIM_1:th.24">SETLIM_1:24</a></i>;<br/><a NAME="E9:105_1_2_5_3"/><b>then </b>
<font color="Maroon" title="c18">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="mesfunc1.html#K15" title="MESFUNC1:func.15">less_dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="mesfunc1.html#K6" title="MESFUNC1:func.6">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c21">n</font></span>)</span></span>)</span>,<font color="Maroon" title="c16">e</font>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E4:105_1_2_5"><i><font color="Green" title="E124">A40</font></i></a></i>;<br/><a NAME="E10:105_1_2_5_3"/><b>then </b>
<font color="Maroon" title="c18">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="mesfunc1.html#K6" title="MESFUNC1:func.6">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c21">n</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="mesfunc1.html#D12" title="MESFUNC1:def.12">MESFUNC1:def 12</a></i>;<br/><b>hence </b><a NAME="E11:105_1_2_5_3"/>
<font color="Maroon" title="c18">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">g</font>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E30:105_1_2"><i><font color="Green" title="E120">A34</font></i></a></i>;<br/></div><b>end;</b></div>
<a NAME="E7:105_1_2_5"/><b>then </b><i><font color="Green" title="E126">A52</font></i>: 
 <a href="kurato_2.html#K4" title="KURATO_2:func.4">lim_inf</a> <font color="Maroon" title="c17">H</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">g</font>
 
<b>by </b><i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/>
<a NAME="E8:105_1_2_5"/>
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">g</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="kurato_2.html#K4" title="KURATO_2:func.4">lim_inf</a> <font color="Maroon" title="c17">H</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="105_1_2_5_4"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="105_1_2_5_4_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c18">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:105_1_2_5_4_1"/><i><font color="Green" title="E127">A53</font></i>: 
<font color="Maroon" title="c18">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">g</font>
 ;<br/><b>then reconsider </b><font color="Maroon" title="c19">x1</font> = <font color="Maroon" title="c18">x</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font> ;<br/><a NAME="E3:105_1_2_5_4_1"/><i><font color="Green" title="E128">A54</font></i>: 
( <font color="Maroon" title="c5">F</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c19">x1</font> is <a href="mesfunc5.html#V10" title="MESFUNC5:attr.10">convergent</a> &amp; <font color="Maroon" title="c4">g</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c18">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="mesfunc5.html#K2" title="MESFUNC5:func.2">lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c19">x1</font></span>)</span> )
 <b>by </b><i><a class="txt" href="mesfunc5.html#E1:105_1_2_5_4_1"><i><font color="Green" title="E127">A53</font></i></a>, <a class="txt" href="mesfunc5.html#E7:105"><i><font color="Green" title="E105">A7</font></i></a></i>;<br/><div><i><font color="Green" title="E129">A55</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="105_1_2_5_4_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:105_1_2_5_4_1_1"/><i><font color="Green" title="E130">A56</font></i>: 
<font color="Maroon" title="c5">F</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c19">x1</font> is <a href="mesfunc5.html#V9" title="MESFUNC5:attr.9">convergent_to_-infty</a>
 ;<br/><b>reconsider </b><font color="Maroon" title="c20">E</font> = <font color="Maroon" title="c16">e</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> <b>by </b><i><a class="txt" href="mesfunc5.html#E1:105_1_2_5"><i><font color="Green" title="E122">A38</font></i></a>, <a class="ref" href="measure5.html#T16" title="MEASURE5:th.16">MEASURE5:16</a></i>;<br/><a NAME="E3:105_1_2_5_4_1_1"/><i><font color="Green" title="E131">A57</font></i>: 
 <a href="real_1.html#K1" title="REAL_1:func.1">-</a> <font color="Maroon" title="c20">E</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 0
 <b>by </b><i><a class="txt" href="mesfunc5.html#E1:105_1_2_5"><i><font color="Green" title="E122">A38</font></i></a>, <a class="ref" href="xreal_1.html#T26" title="XREAL_1:th.26">XREAL_1:26</a></i>;<br/><b>then consider </b><font color="Maroon" title="c21">N</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E5:105_1_2_5_4_1_1"/><i><font color="Green" title="E132">A58</font></i>: 
for <font color="Olive" title="b1">m</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Maroon" title="c21">N</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">m</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c19">x1</font></span>)</span> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Olive" title="b1">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <font color="Maroon" title="c20">E</font></span>)</span>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E1:105_1_2_5_4_1_1"><i><font color="Green" title="E130">A56</font></i></a>, <a class="ref" href="mesfunc5.html#D10" target="_self" title="MESFUNC5:def.10">Def10</a></i>;<br/><a NAME="E6:105_1_2_5_4_1_1"/>
<font color="Maroon" title="c5">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c21">N</font> is <a href="supinf_2.html#V6" title="SUPINF_2:attr.6">nonnegative</a>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E5:105"><i><font color="Green" title="E103">A5</font></i></a></i>;<br/><a NAME="E7:105_1_2_5_4_1_1"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c19">x1</font></span>)</span> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c21">N</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 0 &amp; 0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c21">N</font></span>)</span> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c18">x</font> )
 <b>by </b><i><a class="txt" href="mesfunc5.html#E3:105_1_2_5_4_1_1"><i><font color="Green" title="E131">A57</font></i></a>, <a class="txt" href="mesfunc5.html#E5:105_1_2_5_4_1_1"><i><font color="Green" title="E132">A58</font></i></a>, <a class="ref" href="supinf_2.html#T70" title="SUPINF_2:th.70">SUPINF_2:70</a></i>;<br/><b>hence </b><a NAME="E8:105_1_2_5_4_1_1"/>
contradiction
 <b>by </b><i><a class="ref" href="mesfunc5.html#D13" target="_self" title="MESFUNC5:def.13">Def13</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="105_1_2_5_4_1_2"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="105_1_2_5_4_1_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:105_1_2_5_4_1_2_1"/>
( <font color="Maroon" title="c5">F</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c19">x1</font> is <a href="mesfunc5.html#V7" title="MESFUNC5:attr.7">convergent_to_finite_number</a> or <font color="Maroon" title="c5">F</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c19">x1</font> is <a href="mesfunc5.html#V8" title="MESFUNC5:attr.8">convergent_to_+infty</a> )
 </span><b>by </b><i><a class="txt" href="mesfunc5.html#E4:105_1_2_5_4_1"><i><font color="Green" title="E129">A55</font></i></a>, <a class="txt" href="mesfunc5.html#E3:105_1_2_5_4_1"><i><font color="Green" title="E128">A54</font></i></a>, <a class="ref" href="mesfunc5.html#D11" target="_self" title="MESFUNC5:def.11">Def11</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="105_1_2_5_4_1_2_1_1"><b>suppose </b></a><a NAME="E1:105_1_2_5_4_1_2_1_1"/><i><font color="Green" title="E130">A59</font></i>: 
<font color="Maroon" title="c5">F</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c19">x1</font> is <a href="mesfunc5.html#V7" title="MESFUNC5:attr.7">convergent_to_finite_number</a>
 ;<br/><div class="add"><a NAME="E2:105_1_2_5_4_1_2_1_1"/>
(  ex <font color="Olive" title="b1">limFx</font> being <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>  st <br/>(  <a href="mesfunc5.html#K2" title="MESFUNC5:func.2">lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c19">x1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">limFx</font> &amp; ( for <font color="Olive" title="b2">p</font> being <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>   st 0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Olive" title="b2">p</font> holds <br/> ex <font color="Olive" title="b3">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <br/>for <font color="Olive" title="b4">m</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b3">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b4">m</font> holds <br/><span class="p1"><a href="extreal1.html#K4" title="EXTREAL1:func.4">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c19">x1</font></span>)</span> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Olive" title="b4">m</font></span>)</span> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <span class="p3">(<span class="default"><a href="mesfunc5.html#K2" title="MESFUNC5:func.2">lim</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c19">x1</font></span>)</span></span>)</span></span>)</span></span><a href="extreal1.html#K4" title="EXTREAL1:func.4">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b2">p</font> ) &amp; <font color="Maroon" title="c5">F</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c19">x1</font> is <a href="mesfunc5.html#V7" title="MESFUNC5:attr.7">convergent_to_finite_number</a> ) or (  <a href="mesfunc5.html#K2" title="MESFUNC5:func.2">lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c19">x1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>  &amp; <font color="Maroon" title="c5">F</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c19">x1</font> is <a href="mesfunc5.html#V8" title="MESFUNC5:attr.8">convergent_to_+infty</a> ) or (  <a href="mesfunc5.html#K2" title="MESFUNC5:func.2">lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c19">x1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a>  &amp; <font color="Maroon" title="c5">F</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c19">x1</font> is <a href="mesfunc5.html#V9" title="MESFUNC5:attr.9">convergent_to_-infty</a> ) )
 <b>by </b><i><a class="txt" href="mesfunc5.html#E3:105_1_2_5_4_1"><i><font color="Green" title="E128">A54</font></i></a>, <a class="ref" href="mesfunc5.html#D12" target="_self" title="MESFUNC5:def.12">Def12</a></i>;<br/><b>then consider </b><font color="Maroon" title="c20">limFx</font> being  <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> <b> such that </b><br/><a NAME="E4:105_1_2_5_4_1_2_1_1"/><i><font color="Green" title="E131">A60</font></i>: 
(  <a href="mesfunc5.html#K2" title="MESFUNC5:func.2">lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c19">x1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">limFx</font> &amp; ( for <font color="Olive" title="b1">p</font> being <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>   st 0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Olive" title="b1">p</font> holds <br/> ex <font color="Olive" title="b2">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <br/>for <font color="Olive" title="b3">m</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b2">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b3">m</font> holds <br/><span class="p1"><a href="extreal1.html#K4" title="EXTREAL1:func.4">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c19">x1</font></span>)</span> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Olive" title="b3">m</font></span>)</span> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <span class="p3">(<span class="default"><a href="mesfunc5.html#K2" title="MESFUNC5:func.2">lim</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c19">x1</font></span>)</span></span>)</span></span>)</span></span><a href="extreal1.html#K4" title="EXTREAL1:func.4">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">p</font> ) )
 <b>by </b><i><a class="txt" href="mesfunc5.html#E1:105_1_2_5_4_1_2_1_1"><i><font color="Green" title="E130">A59</font></i></a>, <a class="ref" href="mesfunc5.html#T56" target="_self" title="MESFUNC5:th.56">Th56</a>, <a class="ref" href="mesfunc5.html#T57" target="_self" title="MESFUNC5:th.57">Th57</a></i>;<br/><a NAME="E5:105_1_2_5_4_1_2_1_1"/>
 ex <font color="Olive" title="b1">N</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <br/>for <font color="Olive" title="b2">m</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">N</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">m</font> holds <br/><font color="Maroon" title="c19">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="mesfunc1.html#K15" title="MESFUNC1:func.15">less_dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="mesfunc1.html#K6" title="MESFUNC1:func.6">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Olive" title="b2">m</font></span>)</span></span>)</span>,<font color="Maroon" title="c16">e</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="105_1_2_5_4_1_2_1_1_1"><b>proof </b></a><div class="add">

<b>reconsider </b><font color="Maroon" title="c21">E</font> = <font color="Maroon" title="c16">e</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> <b>by </b><i><a class="txt" href="mesfunc5.html#E1:105_1_2_5"><i><font color="Green" title="E122">A38</font></i></a>, <a class="ref" href="measure5.html#T16" title="MEASURE5:th.16">MEASURE5:16</a></i>;<br/>
<a NAME="E2:105_1_2_5_4_1_2_1_1_1"/><i><font color="Green" title="E132">A61</font></i>: 
0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c21">E</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E1:105_1_2_5"><i><font color="Green" title="E122">A38</font></i></a>, <a class="ref" href="xreal_1.html#T217" title="XREAL_1:th.217">XREAL_1:217</a></i>;<br/>
<b>set </b><font color="Maroon" title="c22">E2</font> = <font color="Maroon" title="c21">E</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2;<br/>
<a NAME="E3:105_1_2_5_4_1_2_1_1_1"/>
<font color="Maroon" title="c4">g</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c18">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="mesfunc5.html#K2" title="MESFUNC5:func.2">lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c19">x1</font></span>)</span>
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E7:105"><i><font color="Green" title="E105">A7</font></i></a>, <a class="txt" href="mesfunc5.html#E1:105_1_2_5_4_1"><i><font color="Green" title="E127">A53</font></i></a></i>;<br/>
<a NAME="E4:105_1_2_5_4_1_2_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c18">x</font></span>)</span> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <span class="p2">(<span class="default"><font color="Maroon" title="c21">E</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="mesfunc5.html#K2" title="MESFUNC5:func.2">lim</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c19">x1</font></span>)</span></span>)</span> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a> 
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E2:105_1_2_5_4_1_2_1_1_1"><i><font color="Green" title="E132">A61</font></i></a>, <a class="ref" href="supinf_2.html#T15" title="SUPINF_2:th.15">SUPINF_2:15</a></i>;<br/>
<a NAME="E5:105_1_2_5_4_1_2_1_1_1"/><b>then </b><i><font color="Green" title="E133">A62</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c18">x</font></span>)</span> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <span class="p2">(<span class="default"><font color="Maroon" title="c21">E</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="mesfunc5.html#K2" title="MESFUNC5:func.2">lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c19">x1</font></span>)</span>
 
<b>by </b><i><a class="ref" href="supinf_2.html#T18" title="SUPINF_2:th.18">SUPINF_2:18</a>, <a class="ref" href="extreal1.html#T24" title="EXTREAL1:th.24">EXTREAL1:24</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c23">N</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E7:105_1_2_5_4_1_2_1_1_1"/><i><font color="Green" title="E134">A63</font></i>: 
for <font color="Olive" title="b1">m</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Maroon" title="c23">N</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">m</font> holds <br/><span class="p1"><a href="extreal1.html#K4" title="EXTREAL1:func.4">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c19">x1</font></span>)</span> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Olive" title="b1">m</font></span>)</span> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <span class="p3">(<span class="default"><a href="mesfunc5.html#K2" title="MESFUNC5:func.2">lim</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c19">x1</font></span>)</span></span>)</span></span>)</span></span><a href="extreal1.html#K4" title="EXTREAL1:func.4">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">E</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E2:105_1_2_5_4_1_2_1_1_1"><i><font color="Green" title="E132">A61</font></i></a>, <a class="txt" href="mesfunc5.html#E4:105_1_2_5_4_1_2_1_1"><i><font color="Green" title="E131">A60</font></i></a></i>;<br/>
<a NAME="E8:105_1_2_5_4_1_2_1_1_1"/><i><font color="Green" title="E135">A64</font></i>: 
for <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="mesfunc5.html#K2" title="MESFUNC5:func.2">lim</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c19">x1</font></span>)</span></span>)</span> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c19">x1</font></span>)</span> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Olive" title="b1">n</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="105_1_2_5_4_1_2_1_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c24">n</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="105_1_2_5_4_1_2_1_1_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c25">e</font> be  <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ;<br/><b>assume </b><a NAME="E1:105_1_2_5_4_1_2_1_1_1_1_1"/>
0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c25">e</font>
 ;<br/><b>then consider </b><font color="Maroon" title="c26">N0</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E3:105_1_2_5_4_1_2_1_1_1_1_1"/><i><font color="Green" title="E136">A65</font></i>: 
for <font color="Olive" title="b1">m</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Maroon" title="c26">N0</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">m</font> holds <br/><span class="p1"><a href="extreal1.html#K4" title="EXTREAL1:func.4">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c19">x1</font></span>)</span> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Olive" title="b1">m</font></span>)</span> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <span class="p3">(<span class="default"><a href="mesfunc5.html#K2" title="MESFUNC5:func.2">lim</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c19">x1</font></span>)</span></span>)</span></span>)</span></span><a href="extreal1.html#K4" title="EXTREAL1:func.4">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c25">e</font>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E4:105_1_2_5_4_1_2_1_1"><i><font color="Green" title="E131">A60</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon" title="c27">N0</font> = <font color="Maroon" title="c26">N0</font>, <font color="Maroon" title="c28">n1</font> = <font color="Maroon" title="c24">n</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/><b>set </b><font color="Maroon" title="c29">m</font> =  <a href="limfunc1.html#K1" title="LIMFUNC1:func.1">max</a> <font color="Maroon" title="c27">N0</font>,<font color="Maroon" title="c28">n1</font>;<br/><a NAME="E5:105_1_2_5_4_1_2_1_1_1_1_1"/>
<font color="Maroon" title="c27">N0</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="limfunc1.html#K1" title="LIMFUNC1:func.1">max</a> <font color="Maroon" title="c27">N0</font>,<font color="Maroon" title="c28">n1</font>
 <b>by </b><i><a class="ref" href="xxreal_0.html#T25" title="XXREAL_0:th.25">XXREAL_0:25</a></i>;<br/><a NAME="E6:105_1_2_5_4_1_2_1_1_1_1_1"/><b>then </b><i><font color="Green" title="E137">A66</font></i>: 
<span class="p1"><a href="extreal1.html#K4" title="EXTREAL1:func.4">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c19">x1</font></span>)</span> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <span class="p4">(<span class="default"><a href="limfunc1.html#K1" title="LIMFUNC1:func.1">max</a> <font color="Maroon" title="c27">N0</font>,<font color="Maroon" title="c28">n1</font></span>)</span></span>)</span> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <span class="p3">(<span class="default"><a href="mesfunc5.html#K2" title="MESFUNC5:func.2">lim</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c19">x1</font></span>)</span></span>)</span></span>)</span></span><a href="extreal1.html#K4" title="EXTREAL1:func.4">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c25">e</font>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E3:105_1_2_5_4_1_2_1_1_1_1_1"><i><font color="Green" title="E136">A65</font></i></a></i>;<br/><a NAME="E7:105_1_2_5_4_1_2_1_1_1_1_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c19">x1</font></span>)</span> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <span class="p2">(<span class="default"><a href="limfunc1.html#K1" title="LIMFUNC1:func.1">max</a> <font color="Maroon" title="c27">N0</font>,<font color="Maroon" title="c28">n1</font></span>)</span></span>)</span> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <span class="p1">(<span class="default"><a href="mesfunc5.html#K2" title="MESFUNC5:func.2">lim</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c19">x1</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="extreal1.html#K4" title="EXTREAL1:func.4">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c19">x1</font></span>)</span> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <span class="p4">(<span class="default"><a href="limfunc1.html#K1" title="LIMFUNC1:func.1">max</a> <font color="Maroon" title="c27">N0</font>,<font color="Maroon" title="c28">n1</font></span>)</span></span>)</span> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <span class="p3">(<span class="default"><a href="mesfunc5.html#K2" title="MESFUNC5:func.2">lim</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c19">x1</font></span>)</span></span>)</span></span>)</span></span><a href="extreal1.html#K4" title="EXTREAL1:func.4">.|</a></span>
 <b>by </b><i><a class="ref" href="extreal2.html#T57" title="EXTREAL2:th.57">EXTREAL2:57</a></i>;<br/><a NAME="E8:105_1_2_5_4_1_2_1_1_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c19">x1</font></span>)</span> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <span class="p2">(<span class="default"><a href="limfunc1.html#K1" title="LIMFUNC1:func.1">max</a> <font color="Maroon" title="c27">N0</font>,<font color="Maroon" title="c28">n1</font></span>)</span></span>)</span> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <span class="p1">(<span class="default"><a href="mesfunc5.html#K2" title="MESFUNC5:func.2">lim</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c19">x1</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c25">e</font>
 <b>by </b><i><a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a>, <a class="txt" href="mesfunc5.html#E6:105_1_2_5_4_1_2_1_1_1_1_1"><i><font color="Green" title="E137">A66</font></i></a></i>;<br/><a NAME="E9:105_1_2_5_4_1_2_1_1_1_1_1"/><b>then </b><i><font color="Green" title="E138">A67</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c19">x1</font></span>)</span> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K1" title="LIMFUNC1:func.1">max</a> <font color="Maroon" title="c27">N0</font>,<font color="Maroon" title="c28">n1</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c25">e</font></span>)</span> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <span class="p1">(<span class="default"><a href="mesfunc5.html#K2" title="MESFUNC5:func.2">lim</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c19">x1</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="extreal2.html#T29" title="EXTREAL2:th.29">EXTREAL2:29</a>, <a class="txt" href="mesfunc5.html#E4:105_1_2_5_4_1_2_1_1"><i><font color="Green" title="E131">A60</font></i></a></i>;<br/><a NAME="E10:105_1_2_5_4_1_2_1_1_1_1_1"/>
<font color="Maroon" title="c24">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="limfunc1.html#K1" title="LIMFUNC1:func.1">max</a> <font color="Maroon" title="c27">N0</font>,<font color="Maroon" title="c28">n1</font>
 <b>by </b><i><a class="ref" href="xxreal_0.html#T25" title="XXREAL_0:th.25">XXREAL_0:25</a></i>;<br/><a NAME="E11:105_1_2_5_4_1_2_1_1_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c24">n</font></span>)</span> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c19">x1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <span class="p2">(<span class="default"><a href="limfunc1.html#K1" title="LIMFUNC1:func.1">max</a> <font color="Maroon" title="c27">N0</font>,<font color="Maroon" title="c28">n1</font></span>)</span></span>)</span> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c19">x1</font>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E6:105"><i><font color="Green" title="E104">A6</font></i></a>, <a class="txt" href="mesfunc5.html#E1:105_1_2_5_4_1"><i><font color="Green" title="E127">A53</font></i></a></i>;<br/><a NAME="E12:105_1_2_5_4_1_2_1_1_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c24">n</font></span>)</span> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c19">x1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c19">x1</font></span>)</span> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K1" title="LIMFUNC1:func.1">max</a> <font color="Maroon" title="c27">N0</font>,<font color="Maroon" title="c28">n1</font></span>)</span>
 <b>by </b><i><a class="ref" href="mesfunc5.html#D13" target="_self" title="MESFUNC5:def.13">Def13</a></i>;<br/><a NAME="E13:105_1_2_5_4_1_2_1_1_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c19">x1</font></span>)</span> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c24">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c19">x1</font></span>)</span> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K1" title="LIMFUNC1:func.1">max</a> <font color="Maroon" title="c27">N0</font>,<font color="Maroon" title="c28">n1</font></span>)</span>
 <b>by </b><i><a class="ref" href="mesfunc5.html#D13" target="_self" title="MESFUNC5:def.13">Def13</a></i>;<br/><b>hence </b><a NAME="E14:105_1_2_5_4_1_2_1_1_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c19">x1</font></span>)</span> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c24">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="mesfunc5.html#K2" title="MESFUNC5:func.2">lim</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c19">x1</font></span>)</span></span>)</span> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c25">e</font></span>)</span>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E9:105_1_2_5_4_1_2_1_1_1_1_1"><i><font color="Green" title="E138">A67</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/></div><b>end;</b></div>
<a NAME="E2:105_1_2_5_4_1_2_1_1_1_1"/><b>then </b><i><font color="Green" title="E136">A68</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c19">x1</font></span>)</span> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c24">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="mesfunc5.html#K2" title="MESFUNC5:func.2">lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c19">x1</font></span>)</span>
 
<b>by </b><i><a class="ref" href="mesfunc5.html#T4" target="_self" title="MESFUNC5:th.4">Th4</a></i>;<br/>
<a NAME="E3:105_1_2_5_4_1_2_1_1_1_1"/>
(  <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="mesfunc5.html#K2" title="MESFUNC5:func.2">lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c19">x1</font></span>)</span> &amp;  <a href="mesfunc5.html#K2" title="MESFUNC5:func.2">lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c19">x1</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>  )
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T4" title="XXREAL_0:th.4">XXREAL_0:4</a>, <a class="ref" href="xxreal_0.html#T5" title="XXREAL_0:th.5">XXREAL_0:5</a></i>;<br/>
<a NAME="E4:105_1_2_5_4_1_2_1_1_1_1"/><b>then </b><i><font color="Green" title="E137">A69</font></i>: 
(  <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="mesfunc5.html#K2" title="MESFUNC5:func.2">lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c19">x1</font></span>)</span> &amp;  <a href="mesfunc5.html#K2" title="MESFUNC5:func.2">lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c19">x1</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>  )
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a>, <a class="txt" href="mesfunc5.html#E4:105_1_2_5_4_1_2_1_1"><i><font color="Green" title="E131">A60</font></i></a></i>;<br/>
<a NAME="E5:105_1_2_5_4_1_2_1_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c19">x1</font></span>)</span> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c24">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a> 
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E2:105_1_2_5_4_1_2_1_1_1_1"><i><font color="Green" title="E136">A68</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<b>hence </b><a NAME="E6:105_1_2_5_4_1_2_1_1_1_1"/>
0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="mesfunc5.html#K2" title="MESFUNC5:func.2">lim</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c19">x1</font></span>)</span></span>)</span> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c19">x1</font></span>)</span> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c24">n</font></span>)</span>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E2:105_1_2_5_4_1_2_1_1_1_1"><i><font color="Green" title="E136">A68</font></i></a>, <a class="txt" href="mesfunc5.html#E4:105_1_2_5_4_1_2_1_1_1_1"><i><font color="Green" title="E137">A69</font></i></a>, <a class="ref" href="extreal1.html#T40" title="EXTREAL1:th.40">EXTREAL1:40</a></i>;<br/>


</div><b>end;</b></div>
<div><i><font color="Green" title="E136">A70</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="105_1_2_5_4_1_2_1_1_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c24">n</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/><a NAME="E1:105_1_2_5_4_1_2_1_1_1_2"/><i><font color="Green" title="E137">A71</font></i>: 
0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="mesfunc5.html#K2" title="MESFUNC5:func.2">lim</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c19">x1</font></span>)</span></span>)</span> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c19">x1</font></span>)</span> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c24">n</font></span>)</span>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E8:105_1_2_5_4_1_2_1_1_1"><i><font color="Green" title="E135">A64</font></i></a></i>;<br/><b>thus </b><span class="p1"><a href="extreal1.html#K4" title="EXTREAL1:func.4">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c19">x1</font></span>)</span> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c24">n</font></span>)</span> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <span class="p3">(<span class="default"><a href="mesfunc5.html#K2" title="MESFUNC5:func.2">lim</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c19">x1</font></span>)</span></span>)</span></span>)</span></span><a href="extreal1.html#K4" title="EXTREAL1:func.4">.|</a></span> = 
<span class="p1"><a href="extreal1.html#K4" title="EXTREAL1:func.4">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="mesfunc5.html#K2" title="MESFUNC5:func.2">lim</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c19">x1</font></span>)</span></span>)</span> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c19">x1</font></span>)</span> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c24">n</font></span>)</span></span>)</span></span><a href="extreal1.html#K4" title="EXTREAL1:func.4">.|</a></span>
<b>by </b><i><a class="ref" href="mesfunc5.html#T1" target="_self" title="MESFUNC5:th.1">Th1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="mesfunc5.html#K2" title="MESFUNC5:func.2">lim</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c19">x1</font></span>)</span></span>)</span> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c19">x1</font></span>)</span> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c24">n</font></span>)</span>
<b>by </b><i><a class="txt" href="mesfunc5.html#E1:105_1_2_5_4_1_2_1_1_1_2"><i><font color="Green" title="E137">A71</font></i></a>, <a class="ref" href="extreal1.html#D3" title="EXTREAL1:def.3">EXTREAL1:def 3</a></i>
;<br/></div><b>end;</b></div>
<div><i><font color="Green" title="E137">A72</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="105_1_2_5_4_1_2_1_1_1_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c24">m</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/><b>assume </b><a NAME="E1:105_1_2_5_4_1_2_1_1_1_3"/><i><font color="Green" title="E138">A73</font></i>: 
<font color="Maroon" title="c23">N</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c24">m</font>
 ;<br/><a NAME="E2:105_1_2_5_4_1_2_1_1_1_3"/><b>then </b>
<span class="p1"><a href="extreal1.html#K4" title="EXTREAL1:func.4">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c19">x1</font></span>)</span> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c24">m</font></span>)</span> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <span class="p3">(<span class="default"><a href="mesfunc5.html#K2" title="MESFUNC5:func.2">lim</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c19">x1</font></span>)</span></span>)</span></span>)</span></span><a href="extreal1.html#K4" title="EXTREAL1:func.4">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">E</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E7:105_1_2_5_4_1_2_1_1_1"><i><font color="Green" title="E134">A63</font></i></a></i>;<br/><a NAME="E3:105_1_2_5_4_1_2_1_1_1_3"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c19">x1</font></span>)</span> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c24">m</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c19">x1</font></span>)</span> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c24">m</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a>  )
 <b>by </b><i><a class="txt" href="mesfunc5.html#E4:105_1_2_5_4_1_2_1_1"><i><font color="Green" title="E131">A60</font></i></a>, <a class="ref" href="mesfunc5.html#T3" target="_self" title="MESFUNC5:th.3">Th3</a></i>;<br/><a NAME="E4:105_1_2_5_4_1_2_1_1_1_3"/><b>then </b><i><font color="Green" title="E139">A74</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c24">m</font></span>)</span> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c18">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c24">m</font></span>)</span> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c18">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a>  )
 <b>by </b><i><a class="ref" href="mesfunc5.html#D13" target="_self" title="MESFUNC5:def.13">Def13</a></i>;<br/><a NAME="E5:105_1_2_5_4_1_2_1_1_1_3"/>
<span class="p1"><a href="extreal1.html#K4" title="EXTREAL1:func.4">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c19">x1</font></span>)</span> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c24">m</font></span>)</span> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <span class="p3">(<span class="default"><a href="mesfunc5.html#K2" title="MESFUNC5:func.2">lim</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c19">x1</font></span>)</span></span>)</span></span>)</span></span><a href="extreal1.html#K4" title="EXTREAL1:func.4">.|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="mesfunc5.html#K2" title="MESFUNC5:func.2">lim</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c19">x1</font></span>)</span></span>)</span> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c19">x1</font></span>)</span> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c24">m</font></span>)</span>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E9:105_1_2_5_4_1_2_1_1_1"><i><font color="Green" title="E136">A70</font></i></a></i>;<br/><a NAME="E6:105_1_2_5_4_1_2_1_1_1_3"/><b>then </b>
<span class="p1">(<span class="default"><a href="mesfunc5.html#K2" title="MESFUNC5:func.2">lim</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c19">x1</font></span>)</span></span>)</span> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c19">x1</font></span>)</span> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c24">m</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">E</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E1:105_1_2_5_4_1_2_1_1_1_3"><i><font color="Green" title="E138">A73</font></i></a>, <a class="txt" href="mesfunc5.html#E7:105_1_2_5_4_1_2_1_1_1"><i><font color="Green" title="E134">A63</font></i></a></i>;<br/><a NAME="E7:105_1_2_5_4_1_2_1_1_1_3"/><b>then </b>
<span class="p1">(<span class="default"><a href="mesfunc5.html#K2" title="MESFUNC5:func.2">lim</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c19">x1</font></span>)</span></span>)</span> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c24">m</font></span>)</span> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c19">x1</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">E</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span>
 <b>by </b><i><a class="ref" href="mesfunc5.html#D13" target="_self" title="MESFUNC5:def.13">Def13</a></i>;<br/><b>hence </b><a NAME="E8:105_1_2_5_4_1_2_1_1_1_3"/>
 <a href="mesfunc5.html#K2" title="MESFUNC5:func.2">lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c19">x1</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c24">m</font></span>)</span> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c18">x</font></span>)</span> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <span class="p2">(<span class="default"><font color="Maroon" title="c21">E</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E4:105_1_2_5_4_1_2_1_1_1_3"><i><font color="Green" title="E139">A74</font></i></a>, <a class="ref" href="extreal2.html#T29" title="EXTREAL2:th.29">EXTREAL2:29</a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="105_1_2_5_4_1_2_1_1_1_4"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c24">m</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/><b>assume </b><a NAME="E1:105_1_2_5_4_1_2_1_1_1_4"/><i><font color="Green" title="E138">A75</font></i>: 
<font color="Maroon" title="c23">N</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c24">m</font>
 ;<br/><a NAME="E2:105_1_2_5_4_1_2_1_1_1_4"/><i><font color="Green" title="E139">A76</font></i>: 
<font color="Maroon" title="c19">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="mesfunc1.html#K6" title="MESFUNC1:func.6">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c24">m</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E30:105_1_2"><i><font color="Green" title="E120">A34</font></i></a>, <a class="txt" href="mesfunc5.html#E1:105_1_2_5_4_1"><i><font color="Green" title="E127">A53</font></i></a></i>;<br/><a NAME="E3:105_1_2_5_4_1_2_1_1_1_4"/><i><font color="Green" title="E140">A77</font></i>: 
<font color="Maroon" title="c4">g</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c18">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="mesfunc5.html#K2" title="MESFUNC5:func.2">lim</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c19">x1</font></span>)</span></span>)</span> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <span class="p2">(<span class="default"><font color="Maroon" title="c21">E</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E5:105_1_2_5_4_1_2_1_1_1"><i><font color="Green" title="E133">A62</font></i></a>, <a class="ref" href="extreal2.html#T33" title="EXTREAL2:th.33">EXTREAL2:33</a></i>;<br/><a NAME="E4:105_1_2_5_4_1_2_1_1_1_4"/>
 <a href="mesfunc5.html#K2" title="MESFUNC5:func.2">lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c19">x1</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c24">m</font></span>)</span> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c18">x</font></span>)</span> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <span class="p2">(<span class="default"><font color="Maroon" title="c21">E</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E10:105_1_2_5_4_1_2_1_1_1"><i><font color="Green" title="E137">A72</font></i></a>, <a class="txt" href="mesfunc5.html#E1:105_1_2_5_4_1_2_1_1_1_4"><i><font color="Green" title="E138">A75</font></i></a></i>;<br/><a NAME="E5:105_1_2_5_4_1_2_1_1_1_4"/><b>then </b>
<span class="p1">(<span class="default"><a href="mesfunc5.html#K2" title="MESFUNC5:func.2">lim</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c19">x1</font></span>)</span></span>)</span> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <span class="p2">(<span class="default"><font color="Maroon" title="c21">E</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c24">m</font></span>)</span> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c18">x</font></span>)</span> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <span class="p2">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <span class="p3">(<span class="default"><font color="Maroon" title="c21">E</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span></span>)</span></span>)</span> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <span class="p2">(<span class="default"><font color="Maroon" title="c21">E</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="mesfunc5.html#T5" target="_self" title="MESFUNC5:th.5">Th5</a></i>;<br/><a NAME="E6:105_1_2_5_4_1_2_1_1_1_4"/><b>then </b>
<font color="Maroon" title="c4">g</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c18">x</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c24">m</font></span>)</span> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c19">x1</font></span>)</span> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <span class="p2">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <span class="p3">(<span class="default"><font color="Maroon" title="c21">E</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span></span>)</span></span>)</span> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <span class="p2">(<span class="default"><font color="Maroon" title="c21">E</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E3:105_1_2_5_4_1_2_1_1_1_4"><i><font color="Green" title="E140">A77</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E7:105_1_2_5_4_1_2_1_1_1_4"/><b>then </b>
<font color="Maroon" title="c4">g</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c18">x</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c24">m</font></span>)</span> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c19">x1</font></span>)</span> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <span class="p3">(<span class="default"><font color="Maroon" title="c21">E</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span></span>)</span> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <span class="p2">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <span class="p3">(<span class="default"><font color="Maroon" title="c21">E</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="extreal1.html#T8" title="EXTREAL1:th.8">EXTREAL1:8</a></i>;<br/><a NAME="E8:105_1_2_5_4_1_2_1_1_1_4"/><b>then </b>
<font color="Maroon" title="c4">g</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c18">x</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c24">m</font></span>)</span> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c19">x1</font></span>)</span> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c21">E</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c21">E</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="mesfunc5.html#T7" target="_self" title="MESFUNC5:th.7">Th7</a></i>;<br/><a NAME="E9:105_1_2_5_4_1_2_1_1_1_4"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c18">x</font></span>)</span> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c24">m</font></span>)</span> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c19">x1</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c16">e</font>
 <b>by </b><i><a class="ref" href="extreal2.html#T30" title="EXTREAL2:th.30">EXTREAL2:30</a></i>;<br/><a NAME="E10:105_1_2_5_4_1_2_1_1_1_4"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="mesfunc1.html#K6" title="MESFUNC1:func.6">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c24">m</font></span>)</span></span>)</span> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c19">x1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c16">e</font>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E2:105_1_2_5_4_1_2_1_1_1_4"><i><font color="Green" title="E139">A76</font></i></a>, <a class="ref" href="mesfunc1.html#D4" title="MESFUNC1:def.4">MESFUNC1:def 4</a></i>;<br/><b>hence </b><a NAME="E11:105_1_2_5_4_1_2_1_1_1_4"/>
<font color="Maroon" title="c19">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="mesfunc1.html#K15" title="MESFUNC1:func.15">less_dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="mesfunc1.html#K6" title="MESFUNC1:func.6">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c24">m</font></span>)</span></span>)</span>,<font color="Maroon" title="c16">e</font>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E2:105_1_2_5_4_1_2_1_1_1_4"><i><font color="Green" title="E139">A76</font></i></a>, <a class="ref" href="mesfunc1.html#D12" title="MESFUNC1:def.12">MESFUNC1:def 12</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E12:105_1_2_5_4_1_2_1_1_1"/>
 ex <font color="Olive" title="b1">N</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <br/>for <font color="Olive" title="b2">m</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">N</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">m</font> holds <br/><font color="Maroon" title="c19">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="mesfunc1.html#K15" title="MESFUNC1:func.15">less_dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="mesfunc1.html#K6" title="MESFUNC1:func.6">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Olive" title="b2">m</font></span>)</span></span>)</span>,<font color="Maroon" title="c16">e</font>
 ;<br/>


</div><b>end;</b></div><b>then consider </b><font color="Maroon" title="c21">N</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E7:105_1_2_5_4_1_2_1_1"/><i><font color="Green" title="E132">A78</font></i>: 
for <font color="Olive" title="b1">m</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Maroon" title="c21">N</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">m</font> holds <br/><font color="Maroon" title="c18">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="mesfunc1.html#K15" title="MESFUNC1:func.15">less_dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="mesfunc1.html#K6" title="MESFUNC1:func.6">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Olive" title="b1">m</font></span>)</span></span>)</span>,<font color="Maroon" title="c16">e</font>
 ;<br/><b>reconsider </b><font color="Maroon" title="c22">N</font> = <font color="Maroon" title="c21">N</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="105_1_2_5_4_1_2_1_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c23">k</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><a NAME="E1:105_1_2_5_4_1_2_1_1_2"/>
<font color="Maroon" title="c22">N</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c23">k</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c22">N</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></i>;<br/><a NAME="E2:105_1_2_5_4_1_2_1_1_2"/><b>then </b>
<font color="Maroon" title="c18">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="mesfunc1.html#K15" title="MESFUNC1:func.15">less_dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="mesfunc1.html#K6" title="MESFUNC1:func.6">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c22">N</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c23">k</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c16">e</font>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E7:105_1_2_5_4_1_2_1_1"><i><font color="Green" title="E132">A78</font></i></a></i>;<br/><b>hence </b><a NAME="E3:105_1_2_5_4_1_2_1_1_2"/>
<font color="Maroon" title="c18">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c17">H</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c22">N</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c23">k</font></span>)</span>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E4:105_1_2_5"><i><font color="Green" title="E124">A40</font></i></a></i>;<br/></div><b>end;</b></div><a NAME="E10:105_1_2_5_4_1_2_1_1"/><b>then </b><i><font color="Green" title="E133">A79</font></i>: 
<font color="Maroon" title="c18">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="setlim_1.html#K1" title="SETLIM_1:func.1">inferior_setsequence</a> <font color="Maroon" title="c17">H</font></span>)</span> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Maroon" title="c22">N</font>
 <b>by </b><i><a class="ref" href="setlim_1.html#T24" title="SETLIM_1:th.24">SETLIM_1:24</a></i>;<br/><a NAME="E11:105_1_2_5_4_1_2_1_1"/>
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="setlim_1.html#K1" title="SETLIM_1:func.1">inferior_setsequence</a> <font color="Maroon" title="c17">H</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 <b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/><b>hence </b><a NAME="E12:105_1_2_5_4_1_2_1_1"/>
 ex <font color="Olive" title="b1">N</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <br/>( <font color="Olive" title="b1">N</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="setlim_1.html#K1" title="SETLIM_1:func.1">inferior_setsequence</a> <font color="Maroon" title="c17">H</font></span>)</span> &amp; <font color="Maroon" title="c18">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="setlim_1.html#K1" title="SETLIM_1:func.1">inferior_setsequence</a> <font color="Maroon" title="c17">H</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">N</font> )
 <b>by </b><i><a class="txt" href="mesfunc5.html#E10:105_1_2_5_4_1_2_1_1"><i><font color="Green" title="E133">A79</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="105_1_2_5_4_1_2_1_2"><b>suppose </b></a><a NAME="E1:105_1_2_5_4_1_2_1_2"/><i><font color="Green" title="E130">A80</font></i>: 
<font color="Maroon" title="c5">F</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c19">x1</font> is <a href="mesfunc5.html#V8" title="MESFUNC5:attr.8">convergent_to_+infty</a>
 ;<br/><div class="add"><a NAME="E2:105_1_2_5_4_1_2_1_2"/>
 ex <font color="Olive" title="b1">N</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <br/>for <font color="Olive" title="b2">m</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">N</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">m</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c19">x1</font></span>)</span> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Olive" title="b2">m</font></span>)</span> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c19">x1</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c16">e</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="105_1_2_5_4_1_2_1_2_1"><b>proof </b></a><div class="add">

<a NAME="E1:105_1_2_5_4_1_2_1_2_1"/>
<font color="Maroon" title="c16">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="supinf_1.html#K3" title="SUPINF_1:func.3">REAL</a> 
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E1:105_1_2_5"><i><font color="Green" title="E122">A38</font></i></a>, <a class="ref" href="measure5.html#T16" title="MEASURE5:th.16">MEASURE5:16</a></i>;<br/>
<a NAME="E2:105_1_2_5_4_1_2_1_2_1"/><b>then </b><i><font color="Green" title="E131">A81</font></i>: 
( <font color="Maroon" title="c16">e</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>  &amp; <font color="Maroon" title="c16">e</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a>  )
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T12" title="XXREAL_0:th.12">XXREAL_0:12</a>, <a class="ref" href="xxreal_0.html#T9" title="XXREAL_0:th.9">XXREAL_0:9</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="105_1_2_5_4_1_2_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:105_1_2_5_4_1_2_1_2_1_1"/>
( <span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c19">x1</font></span>)</span> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <font color="Maroon" title="c16">e</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 0 or 0 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c19">x1</font></span>)</span> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <font color="Maroon" title="c16">e</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="105_1_2_5_4_1_2_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:105_1_2_5_4_1_2_1_2_1_1_1"/><i><font color="Green" title="E132">A82</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c19">x1</font></span>)</span> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <font color="Maroon" title="c16">e</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 0
 ;<br/><div class="add"><b>consider </b><font color="Maroon" title="c20">N</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E3:105_1_2_5_4_1_2_1_2_1_1_1"/><i><font color="Green" title="E133">A83</font></i>: 
for <font color="Olive" title="b1">m</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Maroon" title="c20">N</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">m</font> holds <br/> <a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c19">x1</font></span>)</span> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Olive" title="b1">m</font>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E1:105_1_2_5_4_1_2_1_2"><i><font color="Green" title="E130">A80</font></i></a>, <a class="ref" href="mesfunc5.html#D9" target="_self" title="MESFUNC5:def.9">Def9</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="105_1_2_5_4_1_2_1_2_1_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c21">m</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/><b>assume </b><a NAME="E1:105_1_2_5_4_1_2_1_2_1_1_1_1"/>
<font color="Maroon" title="c20">N</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c21">m</font>
 ;<br/><a NAME="E2:105_1_2_5_4_1_2_1_2_1_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c19">x1</font></span>)</span> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <font color="Maroon" title="c16">e</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c19">x1</font></span>)</span> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c21">m</font>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E1:105_1_2_5_4_1_2_1_2_1_1_1"><i><font color="Green" title="E132">A82</font></i></a>, <a class="txt" href="mesfunc5.html#E3:105_1_2_5_4_1_2_1_2_1_1_1"><i><font color="Green" title="E133">A83</font></i></a></i>;<br/><a NAME="E3:105_1_2_5_4_1_2_1_2_1_1_1_1"/><b>then </b>
<font color="Maroon" title="c4">g</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c19">x1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c19">x1</font></span>)</span> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c21">m</font></span>)</span> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <font color="Maroon" title="c16">e</font>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E2:105_1_2_5_4_1_2_1_2_1"><i><font color="Green" title="E131">A81</font></i></a>, <a class="ref" href="extreal2.html#T29" title="EXTREAL2:th.29">EXTREAL2:29</a></i>;<br/><a NAME="E4:105_1_2_5_4_1_2_1_2_1_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c19">x1</font></span>)</span> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c19">x1</font></span>)</span> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c21">m</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c16">e</font>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E2:105_1_2_5_4_1_2_1_2_1"><i><font color="Green" title="E131">A81</font></i></a>, <a class="ref" href="extreal2.html#T30" title="EXTREAL2:th.30">EXTREAL2:30</a></i>;<br/><b>hence </b><a NAME="E5:105_1_2_5_4_1_2_1_2_1_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c19">x1</font></span>)</span> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c21">m</font></span>)</span> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c19">x1</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c16">e</font>
 <b>by </b><i><a class="ref" href="mesfunc5.html#D13" target="_self" title="MESFUNC5:def.13">Def13</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E5:105_1_2_5_4_1_2_1_2_1_1_1"/>
 ex <font color="Olive" title="b1">N</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <br/>for <font color="Olive" title="b2">m</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">N</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">m</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c19">x1</font></span>)</span> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Olive" title="b2">m</font></span>)</span> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c19">x1</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c16">e</font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="105_1_2_5_4_1_2_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:105_1_2_5_4_1_2_1_2_1_1_2"/><i><font color="Green" title="E132">A84</font></i>: 
0 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c19">x1</font></span>)</span> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <font color="Maroon" title="c16">e</font>
 ;<br/><div class="add"><a NAME="E2:105_1_2_5_4_1_2_1_2_1_1_2"/>
<span class="p1"><a href="extreal1.html#K4" title="EXTREAL1:func.4">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c18">x</font></span>)</span></span><a href="extreal1.html#K4" title="EXTREAL1:func.4">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a> 
 <b>by </b><i><a class="txt" href="mesfunc5.html#E1:105_1_2_5_4_1"><i><font color="Green" title="E127">A53</font></i></a>, <a class="txt" href="mesfunc5.html#E29:105_1_2"><i><font color="Green" title="E119">A33</font></i></a>, <a class="ref" href="mesfunc2.html#D1" title="MESFUNC2:def.1">MESFUNC2:def 1</a></i>;<br/><a NAME="E3:105_1_2_5_4_1_2_1_2_1_1_2"/><b>then </b>
( <font color="Maroon" title="c4">g</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c18">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a>  &amp; <font color="Maroon" title="c4">g</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c18">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>  )
 <b>by </b><i><a class="ref" href="extreal2.html#T67" title="EXTREAL2:th.67">EXTREAL2:67</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c20">gx1</font> = <font color="Maroon" title="c4">g</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c18">x</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> <b>by </b><i><a class="ref" href="xxreal_0.html#T14" title="XXREAL_0:th.14">XXREAL_0:14</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c21">e1</font> = <font color="Maroon" title="c16">e</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> <b>by </b><i><a class="txt" href="mesfunc5.html#E1:105_1_2_5"><i><font color="Green" title="E122">A38</font></i></a>, <a class="ref" href="measure5.html#T16" title="MEASURE5:th.16">MEASURE5:16</a></i>;<br/><a NAME="E6:105_1_2_5_4_1_2_1_2_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c18">x</font></span>)</span> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <font color="Maroon" title="c16">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">gx1</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c21">e1</font>
 <b>by </b><i><a class="ref" href="supinf_2.html#T5" title="SUPINF_2:th.5">SUPINF_2:5</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c22">ee</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c19">x1</font></span>)</span> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <font color="Maroon" title="c16">e</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> ;<br/><a NAME="E8:105_1_2_5_4_1_2_1_2_1_1_2"/><i><font color="Green" title="E133">A85</font></i>: 
( 0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c22">ee</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1 &amp; <font color="Maroon" title="c22">ee</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c22">ee</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1 )
 <b>by </b><i><a class="txt" href="mesfunc5.html#E1:105_1_2_5_4_1_2_1_2_1_1_2"><i><font color="Green" title="E132">A84</font></i></a>, <a class="ref" href="xreal_1.html#T31" title="XREAL_1:th.31">XREAL_1:31</a></i>;<br/><a NAME="E9:105_1_2_5_4_1_2_1_2_1_1_2"/><i><font color="Green" title="E134">A86</font></i>: 
 <a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c22">ee</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <span class="p1">(<span class="default"><font color="Maroon" title="c22">ee</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T31" title="XREAL_1:th.31">XREAL_1:31</a></i>;<br/><b>consider </b><font color="Maroon" title="c23">N</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E11:105_1_2_5_4_1_2_1_2_1_1_2"/><i><font color="Green" title="E135">A87</font></i>: 
for <font color="Olive" title="b1">m</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Maroon" title="c23">N</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">m</font> holds <br/> <a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <span class="p1">(<span class="default"><font color="Maroon" title="c22">ee</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c19">x1</font></span>)</span> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Olive" title="b1">m</font>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E8:105_1_2_5_4_1_2_1_2_1_1_2"><i><font color="Green" title="E133">A85</font></i></a>, <a class="txt" href="mesfunc5.html#E1:105_1_2_5_4_1_2_1_2"><i><font color="Green" title="E130">A80</font></i></a>, <a class="ref" href="mesfunc5.html#D9" target="_self" title="MESFUNC5:def.9">Def9</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="105_1_2_5_4_1_2_1_2_1_1_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c24">m</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/><b>assume </b><a NAME="E1:105_1_2_5_4_1_2_1_2_1_1_2_1"/>
<font color="Maroon" title="c23">N</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c24">m</font>
 ;<br/><a NAME="E2:105_1_2_5_4_1_2_1_2_1_1_2_1"/><b>then </b>
 <a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <span class="p1">(<span class="default"><font color="Maroon" title="c22">ee</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c19">x1</font></span>)</span> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c24">m</font>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E11:105_1_2_5_4_1_2_1_2_1_1_2"><i><font color="Green" title="E135">A87</font></i></a></i>;<br/><a NAME="E3:105_1_2_5_4_1_2_1_2_1_1_2_1"/><b>then </b>
 <a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c22">ee</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c19">x1</font></span>)</span> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c24">m</font>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E9:105_1_2_5_4_1_2_1_2_1_1_2"><i><font color="Green" title="E134">A86</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E4:105_1_2_5_4_1_2_1_2_1_1_2_1"/><b>then </b>
<font color="Maroon" title="c4">g</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c19">x1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c19">x1</font></span>)</span> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c24">m</font></span>)</span> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <font color="Maroon" title="c16">e</font>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E2:105_1_2_5_4_1_2_1_2_1"><i><font color="Green" title="E131">A81</font></i></a>, <a class="ref" href="extreal2.html#T29" title="EXTREAL2:th.29">EXTREAL2:29</a></i>;<br/><a NAME="E5:105_1_2_5_4_1_2_1_2_1_1_2_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c19">x1</font></span>)</span> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c19">x1</font></span>)</span> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c24">m</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c16">e</font>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E2:105_1_2_5_4_1_2_1_2_1"><i><font color="Green" title="E131">A81</font></i></a>, <a class="ref" href="extreal2.html#T30" title="EXTREAL2:th.30">EXTREAL2:30</a></i>;<br/><b>hence </b><a NAME="E6:105_1_2_5_4_1_2_1_2_1_1_2_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c19">x1</font></span>)</span> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c24">m</font></span>)</span> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c19">x1</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c16">e</font>
 <b>by </b><i><a class="ref" href="mesfunc5.html#D13" target="_self" title="MESFUNC5:def.13">Def13</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E13:105_1_2_5_4_1_2_1_2_1_1_2"/>
 ex <font color="Olive" title="b1">N</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <br/>for <font color="Olive" title="b2">m</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">N</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">m</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c19">x1</font></span>)</span> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Olive" title="b2">m</font></span>)</span> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c19">x1</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c16">e</font>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><b>then consider </b><font color="Maroon" title="c20">N</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E4:105_1_2_5_4_1_2_1_2"/><i><font color="Green" title="E131">A88</font></i>: 
for <font color="Olive" title="b1">m</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Maroon" title="c20">N</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">m</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c19">x1</font></span>)</span> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Olive" title="b1">m</font></span>)</span> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c19">x1</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c16">e</font>
 ;<br/><b>reconsider </b><font color="Maroon" title="c21">N</font> = <font color="Maroon" title="c20">N</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/><div><i><font color="Green" title="E132">A89</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="105_1_2_5_4_1_2_1_2_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c22">m</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/><b>assume </b><a NAME="E1:105_1_2_5_4_1_2_1_2_2"/>
<font color="Maroon" title="c21">N</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c22">m</font>
 ;<br/><a NAME="E2:105_1_2_5_4_1_2_1_2_2"/><b>then </b><i><font color="Green" title="E133">A90</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c19">x1</font></span>)</span> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c22">m</font></span>)</span> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c19">x1</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c16">e</font>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E4:105_1_2_5_4_1_2_1_2"><i><font color="Green" title="E131">A88</font></i></a></i>;<br/><a NAME="E3:105_1_2_5_4_1_2_1_2_2"/>
<font color="Maroon" title="c19">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="mesfunc1.html#K6" title="MESFUNC1:func.6">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c22">m</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E30:105_1_2"><i><font color="Green" title="E120">A34</font></i></a>, <a class="txt" href="mesfunc5.html#E1:105_1_2_5_4_1"><i><font color="Green" title="E127">A53</font></i></a></i>;<br/><a NAME="E4:105_1_2_5_4_1_2_1_2_2"/><b>then </b>
( <font color="Maroon" title="c19">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="mesfunc1.html#K6" title="MESFUNC1:func.6">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c22">m</font></span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="mesfunc1.html#K6" title="MESFUNC1:func.6">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c22">m</font></span>)</span></span>)</span> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c19">x1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c16">e</font> )
 <b>by </b><i><a class="txt" href="mesfunc5.html#E2:105_1_2_5_4_1_2_1_2_2"><i><font color="Green" title="E133">A90</font></i></a>, <a class="ref" href="mesfunc1.html#D4" title="MESFUNC1:def.4">MESFUNC1:def 4</a></i>;<br/><b>hence </b><a NAME="E5:105_1_2_5_4_1_2_1_2_2"/>
<font color="Maroon" title="c19">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="mesfunc1.html#K15" title="MESFUNC1:func.15">less_dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="mesfunc1.html#K6" title="MESFUNC1:func.6">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c22">m</font></span>)</span></span>)</span>,<font color="Maroon" title="c16">e</font>
 <b>by </b><i><a class="ref" href="mesfunc1.html#D12" title="MESFUNC1:def.12">MESFUNC1:def 12</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="105_1_2_5_4_1_2_1_2_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c22">k</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><a NAME="E1:105_1_2_5_4_1_2_1_2_3"/>
<font color="Maroon" title="c21">N</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c22">k</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c21">N</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></i>;<br/><a NAME="E2:105_1_2_5_4_1_2_1_2_3"/><b>then </b>
<font color="Maroon" title="c18">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="mesfunc1.html#K15" title="MESFUNC1:func.15">less_dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="mesfunc1.html#K6" title="MESFUNC1:func.6">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c21">N</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c22">k</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c16">e</font>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E6:105_1_2_5_4_1_2_1_2"><i><font color="Green" title="E132">A89</font></i></a></i>;<br/><b>hence </b><a NAME="E3:105_1_2_5_4_1_2_1_2_3"/>
<font color="Maroon" title="c18">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c17">H</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">N</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c22">k</font></span>)</span>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E4:105_1_2_5"><i><font color="Green" title="E124">A40</font></i></a></i>;<br/></div><b>end;</b></div><a NAME="E8:105_1_2_5_4_1_2_1_2"/><b>then </b><i><font color="Green" title="E133">A91</font></i>: 
<font color="Maroon" title="c18">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="setlim_1.html#K1" title="SETLIM_1:func.1">inferior_setsequence</a> <font color="Maroon" title="c17">H</font></span>)</span> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Maroon" title="c21">N</font>
 <b>by </b><i><a class="ref" href="setlim_1.html#T24" title="SETLIM_1:th.24">SETLIM_1:24</a></i>;<br/><a NAME="E9:105_1_2_5_4_1_2_1_2"/>
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="setlim_1.html#K1" title="SETLIM_1:func.1">inferior_setsequence</a> <font color="Maroon" title="c17">H</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 <b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/><b>hence </b><a NAME="E10:105_1_2_5_4_1_2_1_2"/>
 ex <font color="Olive" title="b1">N</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <br/>( <font color="Olive" title="b1">N</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="setlim_1.html#K1" title="SETLIM_1:func.1">inferior_setsequence</a> <font color="Maroon" title="c17">H</font></span>)</span> &amp; <font color="Maroon" title="c18">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="setlim_1.html#K1" title="SETLIM_1:func.1">inferior_setsequence</a> <font color="Maroon" title="c17">H</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">N</font> )
 <b>by </b><i><a class="txt" href="mesfunc5.html#E8:105_1_2_5_4_1_2_1_2"><i><font color="Green" title="E133">A91</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>then consider </b><font color="Maroon" title="c20">N</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E7:105_1_2_5_4_1"/><i><font color="Green" title="E130">A92</font></i>: 
( <font color="Maroon" title="c20">N</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="setlim_1.html#K1" title="SETLIM_1:func.1">inferior_setsequence</a> <font color="Maroon" title="c17">H</font></span>)</span> &amp; <font color="Maroon" title="c18">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="setlim_1.html#K1" title="SETLIM_1:func.1">inferior_setsequence</a> <font color="Maroon" title="c17">H</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">N</font> )
 ;<br/><a NAME="E8:105_1_2_5_4_1"/>
<span class="p1">(<span class="default"><a href="setlim_1.html#K1" title="SETLIM_1:func.1">inferior_setsequence</a> <font color="Maroon" title="c17">H</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">N</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><a href="setlim_1.html#K1" title="SETLIM_1:func.1">inferior_setsequence</a> <font color="Maroon" title="c17">H</font></span>)</span>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E7:105_1_2_5_4_1"><i><font color="Green" title="E130">A92</font></i></a>, <a class="ref" href="funct_1.html#T12" title="FUNCT_1:th.12">FUNCT_1:12</a></i>;<br/><a NAME="E9:105_1_2_5_4_1"/><b>then </b>
<font color="Maroon" title="c18">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="prob_1.html#K2" title="PROB_1:func.2">Union</a> <span class="p1">(<span class="default"><a href="setlim_1.html#K1" title="SETLIM_1:func.1">inferior_setsequence</a> <font color="Maroon" title="c17">H</font></span>)</span>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E7:105_1_2_5_4_1"><i><font color="Green" title="E130">A92</font></i></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/><b>hence </b><a NAME="E10:105_1_2_5_4_1"/>
<font color="Maroon" title="c18">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="kurato_2.html#K4" title="KURATO_2:func.4">lim_inf</a> <font color="Maroon" title="c17">H</font>
 <b>by </b><i><a class="ref" href="setlim_1.html#D4" title="SETLIM_1:def.4">SETLIM_1:def 4</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:105_1_2_5_4"/>
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">g</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="kurato_2.html#K4" title="KURATO_2:func.4">lim_inf</a> <font color="Maroon" title="c17">H</font>
 <b>by </b><i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E9:105_1_2_5"/><b>then </b><i><font color="Green" title="E127">A93</font></i>: 
 <a href="kurato_2.html#K4" title="KURATO_2:func.4">lim_inf</a> <font color="Maroon" title="c17">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">g</font>
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E7:105_1_2_5"><i><font color="Green" title="E126">A52</font></i></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></i>;<br/>
<a NAME="E10:105_1_2_5"/><i><font color="Green" title="E128">A94</font></i>: 
for <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <font color="Maroon" title="c17">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">g</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="105_1_2_5_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c18">n</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="105_1_2_5_5_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c19">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:105_1_2_5_5_1"/>
<font color="Maroon" title="c19">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c17">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">n</font>
 ;<br/><a NAME="E2:105_1_2_5_5_1"/><b>then </b>
<font color="Maroon" title="c19">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="mesfunc1.html#K15" title="MESFUNC1:func.15">less_dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="mesfunc1.html#K6" title="MESFUNC1:func.6">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c18">n</font></span>)</span></span>)</span>,<font color="Maroon" title="c16">e</font>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E4:105_1_2_5"><i><font color="Green" title="E124">A40</font></i></a></i>;<br/><a NAME="E3:105_1_2_5_5_1"/><b>then </b>
<font color="Maroon" title="c19">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="mesfunc1.html#K6" title="MESFUNC1:func.6">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c18">n</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="mesfunc1.html#D12" title="MESFUNC1:def.12">MESFUNC1:def 12</a></i>;<br/><b>hence </b><a NAME="E4:105_1_2_5_5_1"/>
<font color="Maroon" title="c19">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">g</font>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E30:105_1_2"><i><font color="Green" title="E120">A34</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:105_1_2_5_5"/>
<font color="Maroon" title="c17">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">n</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">g</font>
 <b>by </b><i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E11:105_1_2_5"/>
for <font color="Olive" title="b1">n</font>, <font color="Olive" title="b2">m</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">m</font> holds <br/><font color="Maroon" title="c17">H</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Olive" title="b1">n</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c17">H</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Olive" title="b2">m</font>
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E5:105_1_2_5"><i><font color="Green" title="E125">A41</font></i></a></i>;<br/>
<a NAME="E12:105_1_2_5"/><b>then </b><i><font color="Green" title="E129">A95</font></i>: 
<font color="Maroon" title="c17">H</font> is <a href="prob_1.html#V3" title="PROB_1:attr.3">non-decreasing</a>
 
<b>by </b><i><a class="ref" href="prob_1.html#D7" title="PROB_1:def.7">PROB_1:def 7</a></i>;<br/>
<a NAME="E13:105_1_2_5"/><i><font color="Green" title="E130">A96</font></i>: 
 <a href="prob_1.html#K2" title="PROB_1:func.2">Union</a> <font color="Maroon" title="c17">H</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">g</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="105_1_2_5_6"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c18">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:105_1_2_5_6"/>
<font color="Maroon" title="c18">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="prob_1.html#K2" title="PROB_1:func.2">Union</a> <font color="Maroon" title="c17">H</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c19">n</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E3:105_1_2_5_6"/><i><font color="Green" title="E131">A97</font></i>: 
<font color="Maroon" title="c18">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c17">H</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Maroon" title="c19">n</font>
 <b>by </b><i><a class="ref" href="prob_1.html#T25" title="PROB_1:th.25">PROB_1:25</a></i>;<br/>
<a NAME="E4:105_1_2_5_6"/>
<font color="Maroon" title="c17">H</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Maroon" title="c19">n</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">g</font>
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E10:105_1_2_5"><i><font color="Green" title="E128">A94</font></i></a></i>;<br/>
<b>hence </b><a NAME="E5:105_1_2_5_6"/>
<font color="Maroon" title="c18">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">g</font>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E3:105_1_2_5_6"><i><font color="Green" title="E131">A97</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E14:105_1_2_5"/><i><font color="Green" title="E131">A98</font></i>: 
 <a href="kurato_2.html#K4" title="KURATO_2:func.4">lim_inf</a> <font color="Maroon" title="c17">H</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="kurato_2.html#K5" title="KURATO_2:func.5">lim_sup</a> <font color="Maroon" title="c17">H</font>
 
<b>by </b><i><a class="ref" href="kurato_2.html#T9" title="KURATO_2:th.9">KURATO_2:9</a></i>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>2</sub></font>(  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>) <b>-&gt; </b>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K6" title="NUMBERS:func.6">ExtREAL</a>  = <font color="Maroon" title="c3">M</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1</span>)</span>;<br/>
<b>consider </b><font color="Maroon" title="c18">MF</font> being   <a href="mesfunc5.html#NM1" title="MESFUNC5:NM.1">ExtREAL_sequence</a><b> such that </b><br/><a NAME="E16:105_1_2_5"/><i><font color="Green" title="E132">A99</font></i>: 
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon" title="c18">MF</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive" title="b1">n</font>)
 <b>from </b><i><a class="ref" href="funct_2.html#S4" title="FUNCT_2:sch.4">FUNCT_2:sch 4</a>();<br/></i>
<div><i><font color="Green" title="E133">A100</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="105_1_2_5_7"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c19">n</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/><a NAME="E1:105_1_2_5_7"/>
<font color="Maroon" title="c19">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 <b>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/><b>hence </b><a NAME="E2:105_1_2_5_7"/>
<font color="Maroon" title="c18">MF</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c19">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c19">n</font>)
 <b>by </b><i><a class="txt" href="mesfunc5.html#E16:105_1_2_5"><i><font color="Green" title="E132">A99</font></i></a></i>;<br/></div><b>end;</b></div>
<a NAME="E18:105_1_2_5"/><i><font color="Green" title="E134">A101</font></i>: 
( <font color="Maroon" title="c3">M</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">g</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_2.html#K8" title="SUPINF_2:func.8">sup</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K19" title="SUPINF_2:func.19">rng</a> <font color="Maroon" title="c18">MF</font></span>)</span> &amp; ( for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon" title="c17">H</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">S</font> ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="105_1_2_5_8"><b>proof </b></a><div class="add">

<div><i><font color="Green" title="E135">A102</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="105_1_2_5_8_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c19">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:105_1_2_5_8_1"/>
<font color="Maroon" title="c19">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 ;<br/><b>then reconsider </b><font color="Maroon" title="c20">n</font> = <font color="Maroon" title="c19">x</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  ;<br/><a NAME="E3:105_1_2_5_8_1"/><i><font color="Green" title="E136">A103</font></i>: 
<font color="Maroon" title="c5">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c20">n</font> <a href="mesfunc2.html#R1" title="MESFUNC2:pred.1">is_simple_func_in</a> <font color="Maroon" title="c2">S</font>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E3:105"><i><font color="Green" title="E101">A3</font></i></a></i>;<br/><b>then consider </b><font color="Maroon" title="c21">GF</font> being   <a href="mesfunc2.html#NM1" title="MESFUNC2:NM.1">Finite_Sep_Sequence</a> of <font color="Maroon" title="c2">S</font><b> such that </b><br/><a NAME="E5:105_1_2_5_8_1"/><i><font color="Green" title="E137">A104</font></i>: 
(  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c20">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c21">GF</font></span>)</span> &amp; ( for <font color="Olive" title="b1">m</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><br/> for <font color="Olive" title="b2">x</font>, <font color="Olive" title="b3">y</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font>  st <font color="Olive" title="b1">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c21">GF</font> &amp; <font color="Olive" title="b2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c21">GF</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">m</font> &amp; <font color="Olive" title="b3">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c21">GF</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">m</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c20">n</font></span>)</span> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Olive" title="b2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c20">n</font></span>)</span> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Olive" title="b3">y</font> ) )
 <b>by </b><i><a class="ref" href="mesfunc2.html#D5" title="MESFUNC2:def.5">MESFUNC2:def 5</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c22">DGH</font> =  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c21">GF</font></span>)</span> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c2">S</font> <b>by </b><i><a class="ref" href="mesfunc2.html#T34" title="MESFUNC2:th.34">MESFUNC2:34</a></i>;<br/><a NAME="E7:105_1_2_5_8_1"/><i><font color="Green" title="E138">A105</font></i>: 
( <font color="Maroon" title="c4">g</font> is <a href="seq_1.html#V1" title="SEQ_1:attr.1">finite</a> &amp; <font color="Maroon" title="c5">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c20">n</font> is <a href="seq_1.html#V1" title="SEQ_1:attr.1">finite</a> )
 <b>by </b><i><a class="txt" href="mesfunc5.html#E1:105"><i><font color="Green" title="E99">A1</font></i></a>, <a class="txt" href="mesfunc5.html#E3:105_1_2_5_8_1"><i><font color="Green" title="E136">A103</font></i></a>, <a class="ref" href="mesfunc2.html#D5" title="MESFUNC2:def.5">MESFUNC2:def 5</a></i>;<br/><a NAME="E8:105_1_2_5_8_1"/><i><font color="Green" title="E139">A106</font></i>: 
<font color="Maroon" title="c4">g</font> <a href="mesfunc1.html#R2" title="MESFUNC1:pred.2">is_measurable_on</a> <font color="Maroon" title="c22">DGH</font>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E1:105"><i><font color="Green" title="E99">A1</font></i></a>, <a class="ref" href="mesfunc2.html#T37" title="MESFUNC2:th.37">MESFUNC2:37</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c23">E</font> = <font color="Maroon" title="c16">e</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> <b>by </b><i><a class="txt" href="mesfunc5.html#E1:105_1_2_5"><i><font color="Green" title="E122">A38</font></i></a>, <a class="ref" href="measure5.html#T16" title="MEASURE5:th.16">MEASURE5:16</a></i>;<br/><a NAME="E10:105_1_2_5_8_1"/>
<font color="Maroon" title="c5">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c20">n</font> <a href="mesfunc1.html#R2" title="MESFUNC1:pred.2">is_measurable_on</a> <font color="Maroon" title="c22">DGH</font>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E3:105_1_2_5_8_1"><i><font color="Green" title="E136">A103</font></i></a>, <a class="ref" href="mesfunc2.html#T37" title="MESFUNC2:th.37">MESFUNC2:37</a></i>;<br/><a NAME="E11:105_1_2_5_8_1"/><b>then </b>
<font color="Maroon" title="c4">g</font> <a href="mesfunc1.html#K6" title="MESFUNC1:func.6">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c20">n</font></span>)</span> <a href="mesfunc1.html#R2" title="MESFUNC1:pred.2">is_measurable_on</a> <font color="Maroon" title="c22">DGH</font>
 <b>by </b><i><a class="ref" href="mesfunc2.html#T13" title="MESFUNC2:th.13">MESFUNC2:13</a>, <a class="txt" href="mesfunc5.html#E8:105_1_2_5_8_1"><i><font color="Green" title="E139">A106</font></i></a>, <a class="txt" href="mesfunc5.html#E7:105_1_2_5_8_1"><i><font color="Green" title="E138">A105</font></i></a>, <a class="txt" href="mesfunc5.html#E5:105_1_2_5_8_1"><i><font color="Green" title="E137">A104</font></i></a></i>;<br/><a NAME="E12:105_1_2_5_8_1"/><b>then </b>
<font color="Maroon" title="c22">DGH</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K15" title="MESFUNC1:func.15">less_dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="mesfunc1.html#K6" title="MESFUNC1:func.6">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c20">n</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c23">E</font></span>)</span></span>)</span> <a href="mesfunc1.html#R1" title="MESFUNC1:pred.1">is_measurable_on</a> <font color="Maroon" title="c2">S</font>
 <b>by </b><i><a class="ref" href="mesfunc1.html#D17" title="MESFUNC1:def.17">MESFUNC1:def 17</a></i>;<br/><a NAME="E13:105_1_2_5_8_1"/><b>then </b><i><font color="Green" title="E140">A107</font></i>: 
<font color="Maroon" title="c22">DGH</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K15" title="MESFUNC1:func.15">less_dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="mesfunc1.html#K6" title="MESFUNC1:func.6">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c20">n</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c23">E</font></span>)</span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">S</font>
 <b>by </b><i><a class="ref" href="mesfunc1.html#D11" title="MESFUNC1:def.11">MESFUNC1:def 11</a></i>;<br/><a NAME="E14:105_1_2_5_8_1"/><i><font color="Green" title="E141">A108</font></i>: 
 <a href="mesfunc1.html#K15" title="MESFUNC1:func.15">less_dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="mesfunc1.html#K6" title="MESFUNC1:func.6">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c20">n</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c23">E</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="mesfunc1.html#K6" title="MESFUNC1:func.6">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c20">n</font></span>)</span></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="105_1_2_5_8_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c24">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:105_1_2_5_8_1_1"/>
<font color="Maroon" title="c24">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="mesfunc1.html#K15" title="MESFUNC1:func.15">less_dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="mesfunc1.html#K6" title="MESFUNC1:func.6">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c20">n</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c23">E</font></span>)</span>
 ;<br/>

<b>hence </b><a NAME="E2:105_1_2_5_8_1_1"/>
<font color="Maroon" title="c24">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="mesfunc1.html#K6" title="MESFUNC1:func.6">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c20">n</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="mesfunc1.html#D12" title="MESFUNC1:def.12">MESFUNC1:def 12</a></i>;<br/>


</div><b>end;</b></div><a NAME="E15:105_1_2_5_8_1"/>
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c20">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">g</font>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E4:105"><i><font color="Green" title="E102">A4</font></i></a></i>;<br/><a NAME="E16:105_1_2_5_8_1"/><b>then </b>
<font color="Maroon" title="c22">DGH</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K15" title="MESFUNC1:func.15">less_dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="mesfunc1.html#K6" title="MESFUNC1:func.6">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c20">n</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c23">E</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="mesfunc1.html#K6" title="MESFUNC1:func.6">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c20">n</font></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K15" title="MESFUNC1:func.15">less_dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="mesfunc1.html#K6" title="MESFUNC1:func.6">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c20">n</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c23">E</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E30:105_1_2"><i><font color="Green" title="E120">A34</font></i></a>, <a class="txt" href="mesfunc5.html#E5:105_1_2_5_8_1"><i><font color="Green" title="E137">A104</font></i></a></i>;<br/><a NAME="E17:105_1_2_5_8_1"/><b>then </b>
<font color="Maroon" title="c22">DGH</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K15" title="MESFUNC1:func.15">less_dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="mesfunc1.html#K6" title="MESFUNC1:func.6">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c20">n</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c23">E</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mesfunc1.html#K15" title="MESFUNC1:func.15">less_dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="mesfunc1.html#K6" title="MESFUNC1:func.6">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c20">n</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c23">E</font></span>)</span>
 <b>by </b><i><a class="ref" href="xboole_1.html#T28" title="XBOOLE_1:th.28">XBOOLE_1:28</a>, <a class="txt" href="mesfunc5.html#E14:105_1_2_5_8_1"><i><font color="Green" title="E141">A108</font></i></a></i>;<br/><b>hence </b><a NAME="E18:105_1_2_5_8_1"/>
<font color="Maroon" title="c17">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">S</font>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E13:105_1_2_5_8_1"><i><font color="Green" title="E140">A107</font></i></a>, <a class="txt" href="mesfunc5.html#E4:105_1_2_5"><i><font color="Green" title="E124">A40</font></i></a></i>;<br/></div><b>end;</b></div>
<a NAME="E2:105_1_2_5_8"/>
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c17">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c19">HH</font> = <font color="Maroon" title="c17">H</font> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ,<font color="Maroon" title="c2">S</font> <b>by </b><i><a class="txt" href="mesfunc5.html#E1:105_1_2_5_8"><i><font color="Green" title="E135">A102</font></i></a>, <a class="ref" href="funct_2.html#T5" title="FUNCT_2:th.5">FUNCT_2:5</a></i>;<br/>
<div><i><font color="Green" title="E136">A109</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="105_1_2_5_8_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c20">n</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><a NAME="E1:105_1_2_5_8_2"/>
<font color="Maroon" title="c20">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c20">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></i>;<br/><b>hence </b><a NAME="E2:105_1_2_5_8_2"/>
<font color="Maroon" title="c19">HH</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c20">n</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c19">HH</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E5:105_1_2_5"><i><font color="Green" title="E125">A41</font></i></a></i>;<br/></div><b>end;</b></div>
<a NAME="E5:105_1_2_5_8"/>
 <a href="kurato_2.html#K5" title="KURATO_2:func.5">lim_sup</a> <font color="Maroon" title="c17">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="prob_1.html#K2" title="PROB_1:func.2">Union</a> <font color="Maroon" title="c17">H</font>
 
<b>by </b><i><a class="ref" href="setlim_1.html#T65" title="SETLIM_1:th.65">SETLIM_1:65</a>, <a class="txt" href="mesfunc5.html#E12:105_1_2_5"><i><font color="Green" title="E129">A95</font></i></a></i>;<br/>
<a NAME="E6:105_1_2_5_8"/><b>then </b><i><font color="Green" title="E137">A110</font></i>: 
<font color="Maroon" title="c3">M</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <span class="p1">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c17">H</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">M</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">g</font></span>)</span>
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E14:105_1_2_5"><i><font color="Green" title="E131">A98</font></i></a>, <a class="txt" href="mesfunc5.html#E9:105_1_2_5"><i><font color="Green" title="E127">A93</font></i></a>, <a class="txt" href="mesfunc5.html#E13:105_1_2_5"><i><font color="Green" title="E130">A96</font></i></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></i>;<br/>
<a NAME="E7:105_1_2_5_8"/><i><font color="Green" title="E138">A111</font></i>: 
(  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c18">MF</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  &amp;  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c17">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  )
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E8:105_1_2_5_8"/>
 <a href="measure1.html#K5" title="MEASURE1:func.5">rng</a> <font color="Maroon" title="c19">HH</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">S</font>
 
<b>by </b><i><a class="ref" href="relset_1.html#T12" title="RELSET_1:th.12">RELSET_1:12</a></i>;<br/>
<a NAME="E9:105_1_2_5_8"/><b>then </b><i><font color="Green" title="E139">A112</font></i>: 
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c17">H</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c3">M</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E10:105_1_2_5_8"/><i><font color="Green" title="E140">A113</font></i>: 
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds <br/> ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c18">MF</font> iff ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c17">H</font> &amp; <font color="Maroon" title="c17">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c3">M</font> ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="105_1_2_5_8_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c20">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="105_1_2_5_8_3_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:105_1_2_5_8_3_1"/><i><font color="Green" title="E141">A114</font></i>: 
<font color="Maroon" title="c20">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c18">MF</font>
 ;<br/><a NAME="E2:105_1_2_5_8_3_1"/><b>then </b>
<font color="Maroon" title="c17">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c17">H</font>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E7:105_1_2_5_8"><i><font color="Green" title="E138">A111</font></i></a>, <a class="ref" href="funct_1.html#T12" title="FUNCT_1:th.12">FUNCT_1:12</a></i>;<br/><b>hence </b><a NAME="E3:105_1_2_5_8_3_1"/>
( <font color="Maroon" title="c20">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c17">H</font> &amp; <font color="Maroon" title="c17">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c3">M</font> )
 <b>by </b><i><a class="txt" href="mesfunc5.html#E1:105_1_2_5_8_3_1"><i><font color="Green" title="E141">A114</font></i></a>, <a class="txt" href="mesfunc5.html#E9:105_1_2_5_8"><i><font color="Green" title="E139">A112</font></i></a>, <a class="txt" href="mesfunc5.html#E7:105_1_2_5_8"><i><font color="Green" title="E138">A111</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:105_1_2_5_8_3"/>
( <font color="Maroon" title="c20">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c18">MF</font> iff ( <font color="Maroon" title="c20">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c17">H</font> &amp; <font color="Maroon" title="c17">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c3">M</font> ) )
 <b>by </b><i><a class="txt" href="mesfunc5.html#E7:105_1_2_5_8"><i><font color="Green" title="E138">A111</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E11:105_1_2_5_8"/>
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c18">MF</font> holds <br/><font color="Maroon" title="c18">MF</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">M</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font></span>)</span>
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E16:105_1_2_5"><i><font color="Green" title="E132">A99</font></i></a></i>;<br/>
<a NAME="E12:105_1_2_5_8"/><b>then </b>
<font color="Maroon" title="c3">M</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c17">H</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c18">MF</font>
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E10:105_1_2_5_8"><i><font color="Green" title="E140">A113</font></i></a>, <a class="ref" href="funct_1.html#T20" title="FUNCT_1:th.20">FUNCT_1:20</a></i>;<br/>
<b>hence </b><a NAME="E13:105_1_2_5_8"/>
( <font color="Maroon" title="c3">M</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">g</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_2.html#K8" title="SUPINF_2:func.8">sup</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K19" title="SUPINF_2:func.19">rng</a> <font color="Maroon" title="c18">MF</font></span>)</span> &amp; ( for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon" title="c17">H</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">S</font> ) )
 <b>by </b><i><a class="txt" href="mesfunc5.html#E1:105_1_2_5_8"><i><font color="Green" title="E135">A102</font></i></a>, <a class="txt" href="mesfunc5.html#E4:105_1_2_5_8"><i><font color="Green" title="E136">A109</font></i></a>, <a class="txt" href="mesfunc5.html#E6:105_1_2_5_8"><i><font color="Green" title="E137">A110</font></i></a>, <a class="ref" href="measure2.html#T27" title="MEASURE2:th.27">MEASURE2:27</a></i>;<br/>


</div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="105_1_2_5_9"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c19">n</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/><a NAME="E1:105_1_2_5_9"/>
<font color="Maroon" title="c19">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 <b>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/><b>hence </b><a NAME="E2:105_1_2_5_9"/>
<font color="Maroon" title="c17">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">S</font>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E18:105_1_2_5"><i><font color="Green" title="E134">A101</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E20:105_1_2_5"/>
 ex <font color="Olive" title="b1">H</font> being  <a href="prob_1.html#NM2" title="PROB_1:NM.2">SetSequence</a> of <font color="Maroon" title="c1">X</font> ex <font color="Olive" title="b2">MF</font> being  <a href="mesfunc5.html#NM1" title="MESFUNC5:NM.1">ExtREAL_sequence</a> st <br/>( ( for <font color="Olive" title="b3">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <font color="Olive" title="b1">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mesfunc1.html#K15" title="MESFUNC1:func.15">less_dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="mesfunc1.html#K6" title="MESFUNC1:func.6">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Olive" title="b3">n</font></span>)</span></span>)</span>,<font color="Maroon" title="c16">e</font> ) &amp; ( for <font color="Olive" title="b3">n</font>, <font color="Olive" title="b4">m</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b3">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b4">m</font> holds <br/><font color="Olive" title="b1">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">n</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b4">m</font> ) &amp; ( for <font color="Olive" title="b3">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <font color="Olive" title="b1">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">n</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">g</font> ) &amp; ( for <font color="Olive" title="b3">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <font color="Olive" title="b2">MF</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Olive" title="b3">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">M</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">n</font></span>)</span> ) &amp; <font color="Maroon" title="c3">M</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">g</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_2.html#K8" title="SUPINF_2:func.8">sup</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K19" title="SUPINF_2:func.19">rng</a> <font color="Olive" title="b2">MF</font></span>)</span> &amp; ( for <font color="Olive" title="b3">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <font color="Olive" title="b1">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">S</font> ) )
 <b>by </b><i><a class="txt" href="mesfunc5.html#E10:105_1_2_5"><i><font color="Green" title="E128">A94</font></i></a>, <a class="txt" href="mesfunc5.html#E4:105_1_2_5"><i><font color="Green" title="E124">A40</font></i></a>, <a class="txt" href="mesfunc5.html#E5:105_1_2_5"><i><font color="Green" title="E125">A41</font></i></a>, <a class="txt" href="mesfunc5.html#E18:105_1_2_5"><i><font color="Green" title="E134">A101</font></i></a>, <a class="txt" href="mesfunc5.html#E17:105_1_2_5"><i><font color="Green" title="E133">A100</font></i></a></i>;<br/>


</div><b>end;</b></div><a NAME="E32:105_1_2"/><i><font color="Green" title="E122">A115</font></i>: 
( <font color="Maroon" title="c15">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c8">a</font> &amp; <font color="Maroon" title="c8">a</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c15">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">a1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">i</font></span>)</span> )
 <b>by </b><i><a class="txt" href="mesfunc5.html#E8:105_1_2"><i><font color="Green" title="E111">A21</font></i></a>, <a class="txt" href="mesfunc5.html#E22:105_1_2"><i><font color="Green" title="E116">A30</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E33:105_1_2"/><b>then </b><i><font color="Green" title="E123">A116</font></i>: 
0 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <span class="p1">(<span class="default"><a href="seq_4.html#K3" title="SEQ_4:func.3">min</a> <font color="Maroon" title="c10">RINGA</font></span>)</span>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E4:105_1_2"><i><font color="Green" title="E108">A12</font></i></a>, <a class="txt" href="mesfunc5.html#E28:105_1_2"><i><font color="Green" title="E118">A32</font></i></a>, <a class="txt" href="mesfunc5.html#E22:105_1_2"><i><font color="Green" title="E116">A30</font></i></a></i>;<br/><a NAME="E34:105_1_2"/><i><font color="Green" title="E124">A117</font></i>: 
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">g</font> holds <br/>(  <a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <span class="p1">(<span class="default"><a href="seq_4.html#K3" title="SEQ_4:func.3">min</a> <font color="Maroon" title="c10">RINGA</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">g</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Olive" title="b1">x</font> &amp; <font color="Maroon" title="c4">g</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Olive" title="b1">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <span class="p1">(<span class="default"><a href="pre_circ.html#K1" title="PRE_CIRC:func.1">max</a> <font color="Maroon" title="c10">RINGA</font></span>)</span> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="105_1_2_6"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c16">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:105_1_2_6"/><i><font color="Green" title="E125">A118</font></i>: 
<font color="Maroon" title="c16">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">g</font>
 ;<br/>

<a NAME="E2:105_1_2_6"/><b>then </b>
<font color="Maroon" title="c16">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c7">G</font></span>)</span>
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E4:105_1_2"><i><font color="Green" title="E108">A12</font></i></a>, <a class="ref" href="mesfunc3.html#D1" title="MESFUNC3:def.1">MESFUNC3:def 1</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c17">Y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E4:105_1_2_6"/><i><font color="Green" title="E126">A119</font></i>: 
( <font color="Maroon" title="c16">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c17">Y</font> &amp; <font color="Maroon" title="c17">Y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c7">G</font> )
 <b>by </b><i><a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c18">k</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E6:105_1_2_6"/><i><font color="Green" title="E127">A120</font></i>: 
( <font color="Maroon" title="c18">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c7">G</font> &amp; <font color="Maroon" title="c17">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">k</font> )
 <b>by </b><i><a class="txt" href="mesfunc5.html#E4:105_1_2_6"><i><font color="Green" title="E126">A119</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c19">k</font> = <font color="Maroon" title="c18">k</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="txt" href="mesfunc5.html#E6:105_1_2_6"><i><font color="Green" title="E127">A120</font></i></a></i>;<br/>
<a NAME="E8:105_1_2_6"/>
<font color="Maroon" title="c19">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c8">a</font>
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E4:105_1_2"><i><font color="Green" title="E108">A12</font></i></a>, <a class="txt" href="mesfunc5.html#E6:105_1_2_6"><i><font color="Green" title="E127">A120</font></i></a>, <a class="ref" href="mesfunc3.html#D1" title="MESFUNC3:def.1">MESFUNC3:def 1</a></i>;<br/>
<a NAME="E9:105_1_2_6"/><b>then </b><i><font color="Green" title="E128">A121</font></i>: 
<font color="Maroon" title="c19">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">a</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E10:105_1_2_6"/><b>then </b><i><font color="Green" title="E129">A122</font></i>: 
<font color="Maroon" title="c9">a1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c9">a1</font>
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E8:105_1_2"><i><font color="Green" title="E111">A21</font></i></a>, <a class="ref" href="funct_1.html#T12" title="FUNCT_1:th.12">FUNCT_1:12</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="105_1_2_6_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:105_1_2_6_1"/><i><font color="Green" title="E130">A123</font></i>: 
<font color="Maroon" title="c9">a1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">a1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1
 ;<br/><a NAME="E2:105_1_2_6_1"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">a</font>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E9:105_1_2"><i><font color="Green" title="E112">A22</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E3:105_1_2_6_1"/><b>then </b><i><font color="Green" title="E131">A124</font></i>: 
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c9">a1</font>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E8:105_1_2"><i><font color="Green" title="E111">A21</font></i></a></i>;<br/><a NAME="E4:105_1_2_6_1"/>
<font color="Maroon" title="c8">a</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c19">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">a1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">k</font>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E8:105_1_2"><i><font color="Green" title="E111">A21</font></i></a>, <a class="txt" href="mesfunc5.html#E9:105_1_2_6"><i><font color="Green" title="E128">A121</font></i></a></i>;<br/><a NAME="E5:105_1_2_6_1"/><b>then </b><i><font color="Green" title="E132">A125</font></i>: 
<font color="Maroon" title="c8">a</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c19">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">a</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> 1
 <b>by </b><i><a class="txt" href="mesfunc5.html#E3:105_1_2_6_1"><i><font color="Green" title="E131">A124</font></i></a>, <a class="txt" href="mesfunc5.html#E8:105_1_2"><i><font color="Green" title="E111">A21</font></i></a>, <a class="txt" href="mesfunc5.html#E1:105_1_2_6_1"><i><font color="Green" title="E130">A123</font></i></a></i>;<br/><a NAME="E6:105_1_2_6_1"/>
<font color="Maroon" title="c4">g</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c16">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">a</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c19">k</font>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E4:105_1_2_6"><i><font color="Green" title="E126">A119</font></i></a>, <a class="txt" href="mesfunc5.html#E6:105_1_2_6"><i><font color="Green" title="E127">A120</font></i></a>, <a class="txt" href="mesfunc5.html#E4:105_1_2"><i><font color="Green" title="E108">A12</font></i></a>, <a class="ref" href="mesfunc3.html#D1" title="MESFUNC3:def.1">MESFUNC3:def 1</a></i>;<br/><b>hence </b><a NAME="E7:105_1_2_6_1"/>
contradiction
 <b>by </b><i><a class="txt" href="mesfunc5.html#E1:105_1_2_6"><i><font color="Green" title="E125">A118</font></i></a>, <a class="txt" href="mesfunc5.html#E2:105"><i><font color="Green" title="E100">A2</font></i></a>, <a class="txt" href="mesfunc5.html#E5:105_1_2_6_1"><i><font color="Green" title="E132">A125</font></i></a>, <a class="txt" href="mesfunc5.html#E4:105_1_2"><i><font color="Green" title="E108">A12</font></i></a></i>;<br/></div><b>end;</b></div>
<a NAME="E12:105_1_2_6"/><b>then </b>
not <font color="Maroon" title="c9">a1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">a1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<a NAME="E13:105_1_2_6"/><b>then </b><i><font color="Green" title="E130">A126</font></i>: 
<font color="Maroon" title="c9">a1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">RINGA</font>
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E10:105_1_2_6"><i><font color="Green" title="E129">A122</font></i></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/>
<font color="Maroon" title="c4">g</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c16">x</font> = 
<font color="Maroon" title="c8">a</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c19">k</font>
<b>by </b><i><a class="txt" href="mesfunc5.html#E4:105_1_2_6"><i><font color="Green" title="E126">A119</font></i></a>, <a class="txt" href="mesfunc5.html#E6:105_1_2_6"><i><font color="Green" title="E127">A120</font></i></a>, <a class="txt" href="mesfunc5.html#E4:105_1_2"><i><font color="Green" title="E108">A12</font></i></a>, <a class="ref" href="mesfunc3.html#D1" title="MESFUNC3:def.1">MESFUNC3:def 1</a></i>
<br/>.= 
<font color="Maroon" title="c9">a1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">k</font>
<b>by </b><i><a class="txt" href="mesfunc5.html#E8:105_1_2"><i><font color="Green" title="E111">A21</font></i></a>, <a class="txt" href="mesfunc5.html#E9:105_1_2_6"><i><font color="Green" title="E128">A121</font></i></a></i>
;<br/>
<b>hence </b><a NAME="E15:105_1_2_6"/>
(  <a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <span class="p1">(<span class="default"><a href="seq_4.html#K3" title="SEQ_4:func.3">min</a> <font color="Maroon" title="c10">RINGA</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">g</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c16">x</font> &amp; <font color="Maroon" title="c4">g</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c16">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <span class="p1">(<span class="default"><a href="pre_circ.html#K1" title="PRE_CIRC:func.1">max</a> <font color="Maroon" title="c10">RINGA</font></span>)</span> )
 <b>by </b><i><a class="txt" href="mesfunc5.html#E13:105_1_2_6"><i><font color="Green" title="E130">A126</font></i></a>, <a class="ref" href="pre_circ.html#D1" title="PRE_CIRC:def.1">PRE_CIRC:def 1</a>, <a class="ref" href="pre_circ.html#D4" title="PRE_CIRC:def.4">PRE_CIRC:def 4</a></i>;<br/>


</div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="105_1_2_7"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:105_1_2_7"/>
( <font color="Maroon" title="c3">M</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">g</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>  or <font color="Maroon" title="c3">M</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">g</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="105_1_2_7_1"><b>suppose </b></a><a NAME="E1:105_1_2_7_1"/><i><font color="Green" title="E125">A127</font></i>: 
<font color="Maroon" title="c3">M</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">g</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a> 
 ;<br/><div class="add"><b>reconsider </b><font color="Maroon" title="c16">DG</font> =  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">g</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c2">S</font> <b>by </b><i><a class="ref" href="mesfunc5.html#T43" target="_self" title="MESFUNC5:th.43">Th43</a>, <a class="txt" href="mesfunc5.html#E1:105"><i><font color="Green" title="E99">A1</font></i></a></i>;<br/><a NAME="E3:105_1_2_7_1"/><i><font color="Green" title="E126">A128</font></i>: 
<font color="Maroon" title="c3">M</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 <b>by </b><i><a class="ref" href="measure1.html#D11" title="MEASURE1:def.11">MEASURE1:def 11</a></i>;<br/><a NAME="E4:105_1_2_7_1"/>
 <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="measure1.html#R2" title="MEASURE1:pred.2">is_measurable</a> <font color="Maroon" title="c3">M</font>
 <b>by </b><i><a class="ref" href="measure1.html#T66" title="MEASURE1:th.66">MEASURE1:66</a></i>;<br/><a NAME="E5:105_1_2_7_1"/><b>then </b><i><font color="Green" title="E127">A129</font></i>: 
 <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">S</font>
 <b>by </b><i><a class="ref" href="measure1.html#D12" title="MEASURE1:def.12">MEASURE1:def 12</a></i>;<br/><a NAME="E6:105_1_2_7_1"/>
(  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">g</font> &amp;  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">g</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c2">S</font> )
 <b>by </b><i><a class="ref" href="mesfunc5.html#T43" target="_self" title="MESFUNC5:th.43">Th43</a>, <a class="txt" href="mesfunc5.html#E1:105"><i><font color="Green" title="E99">A1</font></i></a>, <a class="ref" href="xboole_1.html#T2" title="XBOOLE_1:th.2">XBOOLE_1:2</a></i>;<br/><a NAME="E7:105_1_2_7_1"/><b>then </b><i><font color="Green" title="E128">A130</font></i>: 
( <font color="Maroon" title="c3">M</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">g</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a>  &amp; <font color="Maroon" title="c3">M</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">g</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>  )
 <b>by </b><i><a class="txt" href="mesfunc5.html#E1:105_1_2_7_1"><i><font color="Green" title="E125">A127</font></i></a>, <a class="txt" href="mesfunc5.html#E3:105_1_2_7_1"><i><font color="Green" title="E126">A128</font></i></a>, <a class="txt" href="mesfunc5.html#E5:105_1_2_7_1"><i><font color="Green" title="E127">A129</font></i></a>, <a class="ref" href="measure1.html#T62" title="MEASURE1:th.62">MEASURE1:62</a></i>;<br/><a NAME="E8:105_1_2_7_1"/>
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">g</font> holds <br/>0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">g</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Olive" title="b1">x</font>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E2:105"><i><font color="Green" title="E100">A2</font></i></a></i>;<br/><a NAME="E9:105_1_2_7_1"/><b>then </b><i><font color="Green" title="E129">A131</font></i>: 
<font color="Maroon" title="c4">g</font> is <a href="supinf_2.html#V6" title="SUPINF_2:attr.6">nonnegative</a>
 <b>by </b><i><a class="ref" href="supinf_2.html#T71" title="SUPINF_2:th.71">SUPINF_2:71</a></i>;<br/><a NAME="E10:105_1_2_7_1"/><i><font color="Green" title="E130">A132</font></i>: 
0 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <span class="p1">(<span class="default"><a href="pre_circ.html#K1" title="PRE_CIRC:func.1">max</a> <font color="Maroon" title="c10">RINGA</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="105_1_2_7_1_1"><b>proof </b></a><div class="add">

<b>consider </b><font color="Maroon" title="c17">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E2:105_1_2_7_1_1"/><i><font color="Green" title="E131">A133</font></i>: 
<font color="Maroon" title="c17">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">g</font>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E1:105_1_2"><i><font color="Green" title="E107">A11</font></i></a>, <a class="ref" href="xboole_0.html#D1" title="XBOOLE_0:def.1">XBOOLE_0:def 1</a></i>;<br/>
<a NAME="E3:105_1_2_7_1_1"/>
(  <a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <span class="p1">(<span class="default"><a href="seq_4.html#K3" title="SEQ_4:func.3">min</a> <font color="Maroon" title="c10">RINGA</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">g</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c17">x</font> &amp; <font color="Maroon" title="c4">g</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c17">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <span class="p1">(<span class="default"><a href="pre_circ.html#K1" title="PRE_CIRC:func.1">max</a> <font color="Maroon" title="c10">RINGA</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E34:105_1_2"><i><font color="Green" title="E124">A117</font></i></a>, <a class="txt" href="mesfunc5.html#E2:105_1_2_7_1_1"><i><font color="Green" title="E131">A133</font></i></a></i>;<br/>
<b>hence </b><a NAME="E4:105_1_2_7_1_1"/>
0 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <span class="p1">(<span class="default"><a href="pre_circ.html#K1" title="PRE_CIRC:func.1">max</a> <font color="Maroon" title="c10">RINGA</font></span>)</span>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E32:105_1_2"><i><font color="Green" title="E122">A115</font></i></a>, <a class="txt" href="mesfunc5.html#E4:105_1_2"><i><font color="Green" title="E108">A12</font></i></a>, <a class="txt" href="mesfunc5.html#E28:105_1_2"><i><font color="Green" title="E118">A32</font></i></a>, <a class="txt" href="mesfunc5.html#E22:105_1_2"><i><font color="Green" title="E116">A30</font></i></a></i>;<br/>


</div><b>end;</b></div><a NAME="E11:105_1_2_7_1"/><i><font color="Green" title="E131">A134</font></i>: 
 <a href="mesfunc5.html#K5" title="MESFUNC5:func.5">integral'</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">g</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <span class="p2">(<span class="default"><a href="pre_circ.html#K1" title="PRE_CIRC:func.1">max</a> <font color="Maroon" title="c10">RINGA</font></span>)</span></span>)</span> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c16">DG</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="105_1_2_7_1_2"><b>proof </b></a><div class="add">

<b>consider </b><font color="Maroon" title="c17">GP</font> being   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <font color="Maroon" title="c1">X</font>, <a href="numbers.html#K6" title="NUMBERS:func.6">ExtREAL</a> <b> such that </b><br/><a NAME="E2:105_1_2_7_1_2"/><i><font color="Green" title="E132">A135</font></i>: 
( <font color="Maroon" title="c17">GP</font> <a href="mesfunc2.html#R1" title="MESFUNC2:pred.1">is_simple_func_in</a> <font color="Maroon" title="c2">S</font> &amp;  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c17">GP</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">DG</font> &amp; ( for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">DG</font> holds <br/><font color="Maroon" title="c17">GP</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <span class="p1">(<span class="default"><a href="pre_circ.html#K1" title="PRE_CIRC:func.1">max</a> <font color="Maroon" title="c10">RINGA</font></span>)</span> ) )
 <b>by </b><i><a class="ref" href="mesfunc5.html#T47" target="_self" title="MESFUNC5:th.47">Th47</a></i>;<br/>
<a NAME="E3:105_1_2_7_1_2"/>
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c17">GP</font> holds <br/>0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c17">GP</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Olive" title="b1">x</font>
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E2:105_1_2_7_1_2"><i><font color="Green" title="E132">A135</font></i></a>, <a class="txt" href="mesfunc5.html#E10:105_1_2_7_1"><i><font color="Green" title="E130">A132</font></i></a></i>;<br/>
<a NAME="E4:105_1_2_7_1_2"/><b>then </b><i><font color="Green" title="E133">A136</font></i>: 
<font color="Maroon" title="c17">GP</font> is <a href="supinf_2.html#V6" title="SUPINF_2:attr.6">nonnegative</a>
 
<b>by </b><i><a class="ref" href="supinf_2.html#T71" title="SUPINF_2:th.71">SUPINF_2:71</a></i>;<br/>
<a NAME="E5:105_1_2_7_1_2"/><i><font color="Green" title="E134">A137</font></i>: 
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">GP</font> <a href="mesfunc1.html#K6" title="MESFUNC1:func.6">-</a> <font color="Maroon" title="c4">g</font></span>)</span> holds <br/><font color="Maroon" title="c4">g</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Olive" title="b1">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c17">GP</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Olive" title="b1">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="105_1_2_7_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c18">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:105_1_2_7_1_2_1"/>
<font color="Maroon" title="c18">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">GP</font> <a href="mesfunc1.html#K6" title="MESFUNC1:func.6">-</a> <font color="Maroon" title="c4">g</font></span>)</span>
 ;<br/>

<a NAME="E2:105_1_2_7_1_2_1"/><b>then </b>
<font color="Maroon" title="c18">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c17">GP</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">g</font></span>)</span></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c17">GP</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p4"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p4"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c17">GP</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p4"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p4"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="mesfunc1.html#D4" title="MESFUNC1:def.4">MESFUNC1:def 4</a></i>;<br/>
<a NAME="E3:105_1_2_7_1_2_1"/><b>then </b><i><font color="Green" title="E135">A138</font></i>: 
<font color="Maroon" title="c18">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c17">GP</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">g</font></span>)</span>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/>
<a NAME="E4:105_1_2_7_1_2_1"/><b>then </b>
<font color="Maroon" title="c17">GP</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c18">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <span class="p1">(<span class="default"><a href="pre_circ.html#K1" title="PRE_CIRC:func.1">max</a> <font color="Maroon" title="c10">RINGA</font></span>)</span>
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E2:105_1_2_7_1_2"><i><font color="Green" title="E132">A135</font></i></a></i>;<br/>
<b>hence </b><a NAME="E5:105_1_2_7_1_2_1"/>
<font color="Maroon" title="c4">g</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c18">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c17">GP</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c18">x</font>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E34:105_1_2"><i><font color="Green" title="E124">A117</font></i></a>, <a class="txt" href="mesfunc5.html#E2:105_1_2_7_1_2"><i><font color="Green" title="E132">A135</font></i></a>, <a class="txt" href="mesfunc5.html#E3:105_1_2_7_1_2_1"><i><font color="Green" title="E135">A138</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E6:105_1_2_7_1_2"/><b>then </b><i><font color="Green" title="E135">A139</font></i>: 
 <a href="mesfunc5.html#K5" title="MESFUNC5:func.5">integral'</a> <font color="Maroon" title="c3">M</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p3">(<span class="default"><font color="Maroon" title="c17">GP</font> <a href="mesfunc1.html#K6" title="MESFUNC1:func.6">-</a> <font color="Maroon" title="c4">g</font></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="mesfunc5.html#K5" title="MESFUNC5:func.5">integral'</a> <font color="Maroon" title="c3">M</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c17">GP</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p3">(<span class="default"><font color="Maroon" title="c17">GP</font> <a href="mesfunc1.html#K6" title="MESFUNC1:func.6">-</a> <font color="Maroon" title="c4">g</font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="mesfunc5.html#T76" target="_self" title="MESFUNC5:th.76">Th76</a>, <a class="txt" href="mesfunc5.html#E1:105"><i><font color="Green" title="E99">A1</font></i></a>, <a class="txt" href="mesfunc5.html#E9:105_1_2_7_1"><i><font color="Green" title="E129">A131</font></i></a>, <a class="txt" href="mesfunc5.html#E4:105_1_2_7_1_2"><i><font color="Green" title="E133">A136</font></i></a>, <a class="txt" href="mesfunc5.html#E2:105_1_2_7_1_2"><i><font color="Green" title="E132">A135</font></i></a></i>;<br/>
<a NAME="E7:105_1_2_7_1_2"/>
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">GP</font> <a href="mesfunc1.html#K6" title="MESFUNC1:func.6">-</a> <font color="Maroon" title="c4">g</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c17">GP</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">g</font></span>)</span>
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E1:105"><i><font color="Green" title="E99">A1</font></i></a>, <a class="txt" href="mesfunc5.html#E2:105_1_2_7_1_2"><i><font color="Green" title="E132">A135</font></i></a>, <a class="txt" href="mesfunc5.html#E4:105_1_2_7_1_2"><i><font color="Green" title="E133">A136</font></i></a>, <a class="txt" href="mesfunc5.html#E9:105_1_2_7_1"><i><font color="Green" title="E129">A131</font></i></a>, <a class="txt" href="mesfunc5.html#E5:105_1_2_7_1_2"><i><font color="Green" title="E134">A137</font></i></a>, <a class="ref" href="mesfunc5.html#T75" target="_self" title="MESFUNC5:th.75">Th75</a></i>;<br/>
<a NAME="E8:105_1_2_7_1_2"/><b>then </b>
( <font color="Maroon" title="c4">g</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c17">GP</font> <a href="mesfunc1.html#K6" title="MESFUNC1:func.6">-</a> <font color="Maroon" title="c4">g</font></span>)</span></span>)</span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c4">g</font> &amp; <font color="Maroon" title="c17">GP</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c17">GP</font> <a href="mesfunc1.html#K6" title="MESFUNC1:func.6">-</a> <font color="Maroon" title="c4">g</font></span>)</span></span>)</span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c17">GP</font> )
 
<b>by </b><i><a class="ref" href="grfunc_1.html#T64" title="GRFUNC_1:th.64">GRFUNC_1:64</a>, <a class="txt" href="mesfunc5.html#E2:105_1_2_7_1_2"><i><font color="Green" title="E132">A135</font></i></a></i>;<br/>
<b>hence </b><a NAME="E9:105_1_2_7_1_2"/>
 <a href="mesfunc5.html#K5" title="MESFUNC5:func.5">integral'</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">g</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <span class="p2">(<span class="default"><a href="pre_circ.html#K1" title="PRE_CIRC:func.1">max</a> <font color="Maroon" title="c10">RINGA</font></span>)</span></span>)</span> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c16">DG</font></span>)</span>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E6:105_1_2_7_1_2"><i><font color="Green" title="E135">A139</font></i></a>, <a class="txt" href="mesfunc5.html#E10:105_1_2_7_1"><i><font color="Green" title="E130">A132</font></i></a>, <a class="txt" href="mesfunc5.html#E2:105_1_2_7_1_2"><i><font color="Green" title="E132">A135</font></i></a>, <a class="ref" href="mesfunc5.html#T77" target="_self" title="MESFUNC5:th.77">Th77</a></i>;<br/>


</div><b>end;</b></div><b>reconsider </b><font color="Maroon" title="c17">MG</font> = <font color="Maroon" title="c3">M</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">g</font></span>)</span> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> <b>by </b><i><a class="txt" href="mesfunc5.html#E7:105_1_2_7_1"><i><font color="Green" title="E128">A130</font></i></a>, <a class="ref" href="xxreal_0.html#T14" title="XXREAL_0:th.14">XXREAL_0:14</a></i>;<br/><a NAME="E13:105_1_2_7_1"/>
<span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <span class="p2">(<span class="default"><a href="pre_circ.html#K1" title="PRE_CIRC:func.1">max</a> <font color="Maroon" title="c10">RINGA</font></span>)</span></span>)</span> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c16">DG</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">beta1</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c17">MG</font>
 <b>by </b><i><a class="ref" href="extreal1.html#T13" title="EXTREAL1:th.13">EXTREAL1:13</a></i>;<br/><a NAME="E14:105_1_2_7_1"/><b>then </b><i><font color="Green" title="E132">A140</font></i>: 
(  <a href="mesfunc5.html#K5" title="MESFUNC5:func.5">integral'</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">g</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>  &amp;  <a href="mesfunc5.html#K5" title="MESFUNC5:func.5">integral'</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">g</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a>  )
 <b>by </b><i><a class="ref" href="mesfunc5.html#T74" target="_self" title="MESFUNC5:th.74">Th74</a>, <a class="txt" href="mesfunc5.html#E1:105"><i><font color="Green" title="E99">A1</font></i></a>, <a class="txt" href="mesfunc5.html#E9:105_1_2_7_1"><i><font color="Green" title="E129">A131</font></i></a>, <a class="txt" href="mesfunc5.html#E11:105_1_2_7_1"><i><font color="Green" title="E131">A134</font></i></a>, <a class="ref" href="xxreal_0.html#T9" title="XXREAL_0:th.9">XXREAL_0:9</a></i>;<br/><a NAME="E15:105_1_2_7_1"/><i><font color="Green" title="E133">A141</font></i>: 
for <font color="Olive" title="b1">e</font> being  <a href="supinf_1.html#NM3" title="SUPINF_1:NM.3">R_eal</a>  st 0 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">e</font> &amp; <font color="Olive" title="b1">e</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <span class="p1">(<span class="default"><a href="seq_4.html#K3" title="SEQ_4:func.3">min</a> <font color="Maroon" title="c10">RINGA</font></span>)</span> holds <br/> ex <font color="Olive" title="b2">N0</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <br/>for <font color="Olive" title="b3">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b2">N0</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b3">n</font> holds <br/><span class="p1">(<span class="default"><a href="mesfunc5.html#K5" title="MESFUNC5:func.5">integral'</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">g</font></span>)</span> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">e</font> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <span class="p4">(<span class="default"><a href="pre_circ.html#K1" title="PRE_CIRC:func.1">max</a> <font color="Maroon" title="c10">RINGA</font></span>)</span></span>)</span> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <span class="p4">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">g</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="mesfunc5.html#K5" title="MESFUNC5:func.5">integral'</a> <font color="Maroon" title="c3">M</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Olive" title="b3">n</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="105_1_2_7_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c18">e</font> be   <a href="supinf_1.html#NM3" title="SUPINF_1:NM.3">R_eal</a>;<br/>

<b>assume </b><a NAME="E1:105_1_2_7_1_3"/><i><font color="Green" title="E134">A142</font></i>: 
( 0 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c18">e</font> &amp; <font color="Maroon" title="c18">e</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <span class="p1">(<span class="default"><a href="seq_4.html#K3" title="SEQ_4:func.3">min</a> <font color="Maroon" title="c10">RINGA</font></span>)</span> )
 ;<br/>

<a NAME="E2:105_1_2_7_1_3"/><b>then </b><i><font color="Green" title="E135">A143</font></i>: 
( <font color="Maroon" title="c18">e</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a>  &amp; <font color="Maroon" title="c18">e</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>  )
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T4" title="XXREAL_0:th.4">XXREAL_0:4</a></i>;<br/>
<div><i><font color="Green" title="E136">A144</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="105_1_2_7_1_3_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c19">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:105_1_2_7_1_3_1"/>
<font color="Maroon" title="c19">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">g</font>
 ;<br/><a NAME="E2:105_1_2_7_1_3_1"/><b>then </b>
<span class="p1"><a href="extreal1.html#K4" title="EXTREAL1:func.4">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c19">x</font></span>)</span></span><a href="extreal1.html#K4" title="EXTREAL1:func.4">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a> 
 <b>by </b><i><a class="txt" href="mesfunc5.html#E29:105_1_2"><i><font color="Green" title="E119">A33</font></i></a>, <a class="ref" href="mesfunc2.html#D1" title="MESFUNC2:def.1">MESFUNC2:def 1</a></i>;<br/><b>hence </b><a NAME="E3:105_1_2_7_1_3_1"/>
( <font color="Maroon" title="c4">g</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c19">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a>  &amp; <font color="Maroon" title="c4">g</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c19">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>  )
 <b>by </b><i><a class="ref" href="extreal2.html#T67" title="EXTREAL2:th.67">EXTREAL2:67</a></i>;<br/></div><b>end;</b></div>
<b>consider </b><font color="Maroon" title="c19">H</font> being   <a href="prob_1.html#NM2" title="PROB_1:NM.2">SetSequence</a> of <font color="Maroon" title="c1">X</font>, <font color="Maroon" title="c20">MF</font> being   <a href="mesfunc5.html#NM1" title="MESFUNC5:NM.1">ExtREAL_sequence</a><b> such that </b><br/><a NAME="E5:105_1_2_7_1_3"/><i><font color="Green" title="E137">A145</font></i>: 
for <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <font color="Maroon" title="c19">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mesfunc1.html#K15" title="MESFUNC1:func.15">less_dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="mesfunc1.html#K6" title="MESFUNC1:func.6">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Olive" title="b1">n</font></span>)</span></span>)</span>,<font color="Maroon" title="c18">e</font>
 <b>and </b><br/><a NAME="E6:105_1_2_7_1_3"/><i><font color="Green" title="E138">A146</font></i>: 
for <font color="Olive" title="b1">n</font>, <font color="Olive" title="b2">m</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">m</font> holds <br/><font color="Maroon" title="c19">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c19">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">m</font>
 <b>and </b><br/><a NAME="E7:105_1_2_7_1_3"/><i><font color="Green" title="E139">A147</font></i>: 
for <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <font color="Maroon" title="c19">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">g</font>
 <b>and </b><br/><a NAME="E8:105_1_2_7_1_3"/><i><font color="Green" title="E140">A148</font></i>: 
for <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <font color="Maroon" title="c20">MF</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">M</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font></span>)</span>
 <b>and </b><br/><a NAME="E9:105_1_2_7_1_3"/><i><font color="Green" title="E141">A149</font></i>: 
<font color="Maroon" title="c3">M</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">g</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_2.html#K8" title="SUPINF_2:func.8">sup</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K19" title="SUPINF_2:func.19">rng</a> <font color="Maroon" title="c20">MF</font></span>)</span>
 <b>and </b><br/><a NAME="E10:105_1_2_7_1_3"/><i><font color="Green" title="E142">A150</font></i>: 
for <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <font color="Maroon" title="c19">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">S</font>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E1:105_1_2_7_1_3"><i><font color="Green" title="E134">A142</font></i></a>, <a class="txt" href="mesfunc5.html#E31:105_1_2"><i><font color="Green" title="E121">A37</font></i></a></i>;<br/>
<a NAME="E11:105_1_2_7_1_3"/>
 <a href="supinf_2.html#K8" title="SUPINF_2:func.8">sup</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K19" title="SUPINF_2:func.19">rng</a> <font color="Maroon" title="c20">MF</font></span>)</span> is   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E7:105_1_2_7_1"><i><font color="Green" title="E128">A130</font></i></a>, <a class="txt" href="mesfunc5.html#E9:105_1_2_7_1_3"><i><font color="Green" title="E141">A149</font></i></a>, <a class="ref" href="xxreal_0.html#T14" title="XXREAL_0:th.14">XXREAL_0:14</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c21">y</font> being  <a href="xxreal_0.html#V1" title="XXREAL_0:attr.1">ext-real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> <b> such that </b><br/><a NAME="E13:105_1_2_7_1_3"/><i><font color="Green" title="E143">A151</font></i>: 
( <font color="Maroon" title="c21">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="supinf_2.html#K19" title="SUPINF_2:func.19">rng</a> <font color="Maroon" title="c20">MF</font> &amp; <span class="p1">(<span class="default"><a href="supinf_2.html#K8" title="SUPINF_2:func.8">sup</a> <span class="p2">(<span class="default"><a href="supinf_2.html#K19" title="SUPINF_2:func.19">rng</a> <font color="Maroon" title="c20">MF</font></span>)</span></span>)</span> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <font color="Maroon" title="c18">e</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c21">y</font> )
 <b>by </b><i><a class="txt" href="mesfunc5.html#E1:105_1_2_7_1_3"><i><font color="Green" title="E134">A142</font></i></a>, <a class="ref" href="measure6.html#T33" title="MEASURE6:th.33">MEASURE6:33</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c22">N0</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E15:105_1_2_7_1_3"/><i><font color="Green" title="E144">A152</font></i>: 
( <font color="Maroon" title="c22">N0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c20">MF</font> &amp; <font color="Maroon" title="c21">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">MF</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c22">N0</font> )
 <b>by </b><i><a class="txt" href="mesfunc5.html#E13:105_1_2_7_1_3"><i><font color="Green" title="E143">A151</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c23">N0</font> = <font color="Maroon" title="c22">N0</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="txt" href="mesfunc5.html#E15:105_1_2_7_1_3"><i><font color="Green" title="E144">A152</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c24">B0</font> = <font color="Maroon" title="c19">H</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Maroon" title="c23">N0</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c2">S</font> <b>by </b><i><a class="txt" href="mesfunc5.html#E10:105_1_2_7_1_3"><i><font color="Green" title="E142">A150</font></i></a></i>;<br/>
<a NAME="E18:105_1_2_7_1_3"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <span class="p2">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">g</font></span>)</span></span>)</span> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <font color="Maroon" title="c18">e</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">M</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">H</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Maroon" title="c23">N0</font></span>)</span>
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E9:105_1_2_7_1_3"><i><font color="Green" title="E141">A149</font></i></a>, <a class="txt" href="mesfunc5.html#E8:105_1_2_7_1_3"><i><font color="Green" title="E140">A148</font></i></a>, <a class="txt" href="mesfunc5.html#E15:105_1_2_7_1_3"><i><font color="Green" title="E144">A152</font></i></a>, <a class="txt" href="mesfunc5.html#E13:105_1_2_7_1_3"><i><font color="Green" title="E143">A151</font></i></a></i>;<br/>
<a NAME="E19:105_1_2_7_1_3"/><b>then </b>
<font color="Maroon" title="c3">M</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">g</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c19">H</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Maroon" title="c23">N0</font></span>)</span></span>)</span> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <font color="Maroon" title="c18">e</font>
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E2:105_1_2_7_1_3"><i><font color="Green" title="E135">A143</font></i></a>, <a class="ref" href="extreal2.html#T29" title="EXTREAL2:th.29">EXTREAL2:29</a></i>;<br/>
<a NAME="E20:105_1_2_7_1_3"/><b>then </b><i><font color="Green" title="E145">A153</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <span class="p2">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">g</font></span>)</span></span>)</span> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c19">H</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Maroon" title="c23">N0</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c18">e</font>
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E2:105_1_2_7_1_3"><i><font color="Green" title="E135">A143</font></i></a>, <a class="ref" href="extreal2.html#T30" title="EXTREAL2:th.30">EXTREAL2:30</a></i>;<br/>
<a NAME="E21:105_1_2_7_1_3"/><i><font color="Green" title="E146">A154</font></i>: 
<font color="Maroon" title="c24">B0</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c16">DG</font>
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E7:105_1_2_7_1_3"><i><font color="Green" title="E139">A147</font></i></a></i>;<br/>
<a NAME="E22:105_1_2_7_1_3"/><b>then </b><i><font color="Green" title="E147">A155</font></i>: 
<font color="Maroon" title="c3">M</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c24">B0</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">M</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c16">DG</font>
 
<b>by </b><i><a class="ref" href="measure1.html#T62" title="MEASURE1:th.62">MEASURE1:62</a></i>;<br/>
<a NAME="E23:105_1_2_7_1_3"/>
<font color="Maroon" title="c3">M</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c16">DG</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a> 
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E1:105_1_2_7_1"><i><font color="Green" title="E125">A127</font></i></a>, <a class="ref" href="xxreal_0.html#T4" title="XXREAL_0:th.4">XXREAL_0:4</a></i>;<br/>
<a NAME="E24:105_1_2_7_1_3"/><b>then </b>
<font color="Maroon" title="c3">M</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c24">B0</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a> 
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E22:105_1_2_7_1_3"><i><font color="Green" title="E147">A155</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E25:105_1_2_7_1_3"/><b>then </b><i><font color="Green" title="E148">A156</font></i>: 
<font color="Maroon" title="c3">M</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">DG</font> <a href="measure1.html#K4" title="MEASURE1:func.4">\</a> <font color="Maroon" title="c24">B0</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c16">DG</font></span>)</span> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c24">B0</font></span>)</span>
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E21:105_1_2_7_1_3"><i><font color="Green" title="E146">A154</font></i></a>, <a class="ref" href="measure1.html#T63" title="MEASURE1:th.63">MEASURE1:63</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c23">N0</font>
;<br/>

<div><i><font color="Green" title="E149">A157</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="105_1_2_7_1_3_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c25">n</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/><b>assume </b><a NAME="E1:105_1_2_7_1_3_2"/><i><font color="Green" title="E150">A158</font></i>: 
<font color="Maroon" title="c23">N0</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c25">n</font>
 ;<br/><b>reconsider </b><font color="Maroon" title="c26">BN</font> = <font color="Maroon" title="c19">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">n</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c2">S</font> <b>by </b><i><a class="txt" href="mesfunc5.html#E10:105_1_2_7_1_3"><i><font color="Green" title="E142">A150</font></i></a></i>;<br/><a NAME="E3:105_1_2_7_1_3_2"/>
<font color="Maroon" title="c19">H</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Maroon" title="c23">N0</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c19">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">n</font>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E6:105_1_2_7_1_3"><i><font color="Green" title="E138">A146</font></i></a>, <a class="txt" href="mesfunc5.html#E1:105_1_2_7_1_3_2"><i><font color="Green" title="E150">A158</font></i></a></i>;<br/><a NAME="E4:105_1_2_7_1_3_2"/><b>then </b>
<font color="Maroon" title="c16">DG</font> <a href="measure1.html#K4" title="MEASURE1:func.4">\</a> <font color="Maroon" title="c26">BN</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c16">DG</font> <a href="measure1.html#K4" title="MEASURE1:func.4">\</a> <font color="Maroon" title="c24">B0</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T34" title="XBOOLE_1:th.34">XBOOLE_1:34</a></i>;<br/><a NAME="E5:105_1_2_7_1_3_2"/><b>then </b>
<font color="Maroon" title="c3">M</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">DG</font> <a href="measure1.html#K4" title="MEASURE1:func.4">\</a> <font color="Maroon" title="c26">BN</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">M</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">DG</font> <a href="measure1.html#K4" title="MEASURE1:func.4">\</a> <font color="Maroon" title="c24">B0</font></span>)</span>
 <b>by </b><i><a class="ref" href="measure1.html#T62" title="MEASURE1:th.62">MEASURE1:62</a></i>;<br/><b>hence </b><a NAME="E6:105_1_2_7_1_3_2"/>
<font color="Maroon" title="c3">M</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">g</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p2">(<span class="default"><font color="Maroon" title="c19">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">n</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c18">e</font>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E20:105_1_2_7_1_3"><i><font color="Green" title="E145">A153</font></i></a>, <a class="txt" href="mesfunc5.html#E25:105_1_2_7_1_3"><i><font color="Green" title="E148">A156</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="105_1_2_7_1_3_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c25">n</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/><b>assume </b><a NAME="E1:105_1_2_7_1_3_3"/><i><font color="Green" title="E150">A159</font></i>: 
<font color="Maroon" title="c23">N0</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c25">n</font>
 ;<br/><a NAME="E2:105_1_2_7_1_3_3"/><i><font color="Green" title="E151">A160</font></i>: 
<font color="Maroon" title="c19">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">n</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c16">DG</font>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E7:105_1_2_7_1_3"><i><font color="Green" title="E139">A147</font></i></a></i>;<br/><a NAME="E3:105_1_2_7_1_3_3"/><b>then </b>
<font color="Maroon" title="c16">DG</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">DG</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">n</font></span>)</span>
 <b>by </b><i><a class="ref" href="xboole_1.html#T12" title="XBOOLE_1:th.12">XBOOLE_1:12</a></i>;<br/><a NAME="E4:105_1_2_7_1_3_3"/><b>then </b><i><font color="Green" title="E152">A161</font></i>: 
<font color="Maroon" title="c16">DG</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">DG</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p2">(<span class="default"><font color="Maroon" title="c19">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">n</font></span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">n</font></span>)</span>
 <b>by </b><i><a class="ref" href="xboole_1.html#T39" title="XBOOLE_1:th.39">XBOOLE_1:39</a></i>;<br/><a NAME="E5:105_1_2_7_1_3_3"/>
<font color="Maroon" title="c19">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">n</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c2">S</font>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E10:105_1_2_7_1_3"><i><font color="Green" title="E142">A150</font></i></a></i>;<br/><a NAME="E6:105_1_2_7_1_3_3"/><b>then </b>
<font color="Maroon" title="c19">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">n</font> <a href="measure1.html#R2" title="MEASURE1:pred.2">is_measurable</a> <font color="Maroon" title="c3">M</font>
 <b>by </b><i><a class="ref" href="measure1.html#D12" title="MEASURE1:def.12">MEASURE1:def 12</a></i>;<br/><a NAME="E7:105_1_2_7_1_3_3"/><b>then </b><i><font color="Green" title="E153">A162</font></i>: 
<font color="Maroon" title="c1">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">n</font></span>)</span> <a href="measure1.html#R2" title="MEASURE1:pred.2">is_measurable</a> <font color="Maroon" title="c3">M</font>
 <b>by </b><i><a class="ref" href="measure1.html#T66" title="MEASURE1:th.66">MEASURE1:66</a></i>;<br/><a NAME="E8:105_1_2_7_1_3_3"/>
<font color="Maroon" title="c16">DG</font> <a href="measure1.html#R2" title="MEASURE1:pred.2">is_measurable</a> <font color="Maroon" title="c3">M</font>
 <b>by </b><i><a class="ref" href="measure1.html#D12" title="MEASURE1:def.12">MEASURE1:def 12</a></i>;<br/><a NAME="E9:105_1_2_7_1_3_3"/><b>then </b><i><font color="Green" title="E154">A163</font></i>: 
<font color="Maroon" title="c16">DG</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2">(<span class="default"><font color="Maroon" title="c19">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">n</font></span>)</span></span>)</span> <a href="measure1.html#R2" title="MEASURE1:pred.2">is_measurable</a> <font color="Maroon" title="c3">M</font>
 <b>by </b><i><a class="ref" href="measure1.html#T66" title="MEASURE1:th.66">MEASURE1:66</a>, <a class="txt" href="mesfunc5.html#E7:105_1_2_7_1_3_3"><i><font color="Green" title="E153">A162</font></i></a></i>;<br/><a NAME="E10:105_1_2_7_1_3_3"/>
<font color="Maroon" title="c16">DG</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2">(<span class="default"><font color="Maroon" title="c19">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">n</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">DG</font> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">n</font></span>)</span>
 <b>by </b><i><a class="ref" href="xboole_1.html#T49" title="XBOOLE_1:th.49">XBOOLE_1:49</a></i>;<br/><a NAME="E11:105_1_2_7_1_3_3"/><b>then </b>
<font color="Maroon" title="c16">DG</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2">(<span class="default"><font color="Maroon" title="c19">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">n</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">DG</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">n</font></span>)</span>
 <b>by </b><i><a class="ref" href="xboole_1.html#T28" title="XBOOLE_1:th.28">XBOOLE_1:28</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c26">A</font> = <font color="Maroon" title="c16">DG</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">n</font></span>)</span> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c2">S</font> <b>by </b><i><a class="txt" href="mesfunc5.html#E9:105_1_2_7_1_3_3"><i><font color="Green" title="E154">A163</font></i></a>, <a class="ref" href="measure1.html#D12" title="MEASURE1:def.12">MEASURE1:def 12</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c27">B</font> = <font color="Maroon" title="c19">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">n</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c2">S</font> <b>by </b><i><a class="txt" href="mesfunc5.html#E10:105_1_2_7_1_3"><i><font color="Green" title="E142">A150</font></i></a></i>;<br/><a NAME="E14:105_1_2_7_1_3_3"/><i><font color="Green" title="E155">A164</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c25">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">g</font>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E4:105"><i><font color="Green" title="E102">A4</font></i></a></i>;<br/><a NAME="E15:105_1_2_7_1_3_3"/><b>then </b><i><font color="Green" title="E156">A165</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c25">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c26">A</font> <a href="measure1.html#K2" title="MEASURE1:func.2">\/</a> <font color="Maroon" title="c27">B</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c25">n</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E4:105_1_2_7_1_3_3"><i><font color="Green" title="E152">A161</font></i></a></i>;<br/><a NAME="E16:105_1_2_7_1_3_3"/>
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c25">n</font></span>)</span> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c25">n</font></span>)</span> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c25">n</font></span>)</span> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Olive" title="b1">x</font>
 ;<br/><a NAME="E17:105_1_2_7_1_3_3"/><b>then </b><i><font color="Green" title="E157">A166</font></i>: 
<font color="Maroon" title="c5">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c25">n</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c25">n</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c26">A</font> <a href="measure1.html#K2" title="MEASURE1:func.2">\/</a> <font color="Maroon" title="c27">B</font></span>)</span>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E15:105_1_2_7_1_3_3"><i><font color="Green" title="E156">A165</font></i></a>, <a class="ref" href="funct_1.html#T68" title="FUNCT_1:th.68">FUNCT_1:68</a></i>;<br/><a NAME="E18:105_1_2_7_1_3_3"/><i><font color="Green" title="E158">A167</font></i>: 
<font color="Maroon" title="c5">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c25">n</font> is <a href="supinf_2.html#V6" title="SUPINF_2:attr.6">nonnegative</a>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E5:105"><i><font color="Green" title="E103">A5</font></i></a></i>;<br/><a NAME="E19:105_1_2_7_1_3_3"/><b>then </b><i><font color="Green" title="E159">A168</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c25">n</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c26">A</font> is <a href="supinf_2.html#V6" title="SUPINF_2:attr.6">nonnegative</a> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c25">n</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c27">B</font> is <a href="supinf_2.html#V6" title="SUPINF_2:attr.6">nonnegative</a> )
 <b>by </b><i><a class="ref" href="mesfunc5.html#T21" target="_self" title="MESFUNC5:th.21">Th21</a></i>;<br/><a NAME="E20:105_1_2_7_1_3_3"/><i><font color="Green" title="E160">A169</font></i>: 
<font color="Maroon" title="c26">A</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c27">B</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T79" title="XBOOLE_1:th.79">XBOOLE_1:79</a></i>;<br/><a NAME="E21:105_1_2_7_1_3_3"/><i><font color="Green" title="E161">A170</font></i>: 
<font color="Maroon" title="c5">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c25">n</font> <a href="mesfunc2.html#R1" title="MESFUNC2:pred.1">is_simple_func_in</a> <font color="Maroon" title="c2">S</font>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E3:105"><i><font color="Green" title="E101">A3</font></i></a></i>;<br/><a NAME="E22:105_1_2_7_1_3_3"/><b>then </b><i><font color="Green" title="E162">A171</font></i>: 
 <a href="mesfunc5.html#K5" title="MESFUNC5:func.5">integral'</a> <font color="Maroon" title="c3">M</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c25">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="mesfunc5.html#K5" title="MESFUNC5:func.5">integral'</a> <font color="Maroon" title="c3">M</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c25">n</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c26">A</font></span>)</span></span>)</span> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <span class="p1">(<span class="default"><a href="mesfunc5.html#K5" title="MESFUNC5:func.5">integral'</a> <font color="Maroon" title="c3">M</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c25">n</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c27">B</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E18:105_1_2_7_1_3_3"><i><font color="Green" title="E158">A167</font></i></a>, <a class="txt" href="mesfunc5.html#E20:105_1_2_7_1_3_3"><i><font color="Green" title="E160">A169</font></i></a>, <a class="ref" href="mesfunc5.html#T73" target="_self" title="MESFUNC5:th.73">Th73</a>, <a class="txt" href="mesfunc5.html#E17:105_1_2_7_1_3_3"><i><font color="Green" title="E157">A166</font></i></a></i>;<br/><a NAME="E23:105_1_2_7_1_3_3"/><i><font color="Green" title="E163">A172</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c25">n</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c26">A</font> <a href="mesfunc2.html#R1" title="MESFUNC2:pred.1">is_simple_func_in</a> <font color="Maroon" title="c2">S</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c25">n</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c27">B</font> <a href="mesfunc2.html#R1" title="MESFUNC2:pred.1">is_simple_func_in</a> <font color="Maroon" title="c2">S</font> )
 <b>by </b><i><a class="txt" href="mesfunc5.html#E21:105_1_2_7_1_3_3"><i><font color="Green" title="E161">A170</font></i></a>, <a class="ref" href="mesfunc5.html#T40" target="_self" title="MESFUNC5:th.40">Th40</a></i>;<br/><a NAME="E24:105_1_2_7_1_3_3"/><b>then </b>
( 0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="mesfunc5.html#K5" title="MESFUNC5:func.5">integral'</a> <font color="Maroon" title="c3">M</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c25">n</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c26">A</font></span>)</span> &amp; 0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="mesfunc5.html#K5" title="MESFUNC5:func.5">integral'</a> <font color="Maroon" title="c3">M</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c25">n</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c27">B</font></span>)</span> )
 <b>by </b><i><a class="ref" href="mesfunc5.html#T74" target="_self" title="MESFUNC5:th.74">Th74</a>, <a class="txt" href="mesfunc5.html#E19:105_1_2_7_1_3_3"><i><font color="Green" title="E159">A168</font></i></a></i>;<br/><a NAME="E25:105_1_2_7_1_3_3"/><b>then </b><i><font color="Green" title="E164">A173</font></i>: 
 <a href="mesfunc5.html#K5" title="MESFUNC5:func.5">integral'</a> <font color="Maroon" title="c3">M</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c25">n</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c27">B</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="mesfunc5.html#K5" title="MESFUNC5:func.5">integral'</a> <font color="Maroon" title="c3">M</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c25">n</font></span>)</span>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E22:105_1_2_7_1_3_3"><i><font color="Green" title="E162">A171</font></i></a>, <a class="ref" href="supinf_2.html#T20" title="SUPINF_2:th.20">SUPINF_2:20</a></i>;<br/><a NAME="E26:105_1_2_7_1_3_3"/>
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c25">n</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c27">B</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c25">n</font></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c27">B</font>
 <b>by </b><i><a class="ref" href="funct_1.html#T68" title="FUNCT_1:th.68">FUNCT_1:68</a></i>;<br/><a NAME="E27:105_1_2_7_1_3_3"/><b>then </b><i><font color="Green" title="E165">A174</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c25">n</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c27">B</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c27">B</font>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E14:105_1_2_7_1_3_3"><i><font color="Green" title="E155">A164</font></i></a>, <a class="txt" href="mesfunc5.html#E2:105_1_2_7_1_3_3"><i><font color="Green" title="E151">A160</font></i></a>, <a class="ref" href="xboole_1.html#T28" title="XBOOLE_1:th.28">XBOOLE_1:28</a></i>;<br/><b>consider </b><font color="Maroon" title="c28">EP</font> being   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <font color="Maroon" title="c1">X</font>, <a href="numbers.html#K6" title="NUMBERS:func.6">ExtREAL</a> <b> such that </b><br/><a NAME="E29:105_1_2_7_1_3_3"/><i><font color="Green" title="E166">A175</font></i>: 
( <font color="Maroon" title="c28">EP</font> <a href="mesfunc2.html#R1" title="MESFUNC2:pred.1">is_simple_func_in</a> <font color="Maroon" title="c2">S</font> &amp;  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c28">EP</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c27">B</font> &amp; ( for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c27">B</font> holds <br/><font color="Maroon" title="c28">EP</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">e</font> ) )
 <b>by </b><i><a class="txt" href="mesfunc5.html#E2:105_1_2_7_1_3"><i><font color="Green" title="E135">A143</font></i></a>, <a class="ref" href="mesfunc5.html#T47" target="_self" title="MESFUNC5:th.47">Th47</a></i>;<br/><a NAME="E30:105_1_2_7_1_3_3"/><i><font color="Green" title="E167">A176</font></i>: 
 <a href="mesfunc5.html#K5" title="MESFUNC5:func.5">integral'</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c28">EP</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">e</font> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c27">B</font></span>)</span>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E29:105_1_2_7_1_3_3"><i><font color="Green" title="E166">A175</font></i></a>, <a class="txt" href="mesfunc5.html#E1:105_1_2_7_1_3"><i><font color="Green" title="E134">A142</font></i></a>, <a class="ref" href="mesfunc5.html#T77" target="_self" title="MESFUNC5:th.77">Th77</a></i>;<br/><a NAME="E31:105_1_2_7_1_3_3"/><i><font color="Green" title="E168">A177</font></i>: 
<font color="Maroon" title="c3">M</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">g</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a> 
 <b>by </b><i><a class="txt" href="mesfunc5.html#E1:105_1_2_7_1"><i><font color="Green" title="E125">A127</font></i></a>, <a class="ref" href="xxreal_0.html#T4" title="XXREAL_0:th.4">XXREAL_0:4</a></i>;<br/><a NAME="E32:105_1_2_7_1_3_3"/><i><font color="Green" title="E169">A178</font></i>: 
 <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c27">B</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T2" title="XBOOLE_1:th.2">XBOOLE_1:2</a></i>;<br/><a NAME="E33:105_1_2_7_1_3_3"/><b>then </b><i><font color="Green" title="E170">A179</font></i>: 
0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">M</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c27">B</font>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E3:105_1_2_7_1"><i><font color="Green" title="E126">A128</font></i></a>, <a class="txt" href="mesfunc5.html#E5:105_1_2_7_1"><i><font color="Green" title="E127">A129</font></i></a>, <a class="ref" href="measure1.html#T62" title="MEASURE1:th.62">MEASURE1:62</a></i>;<br/><a NAME="E34:105_1_2_7_1_3_3"/><i><font color="Green" title="E171">A180</font></i>: 
<font color="Maroon" title="c3">M</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c27">B</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">M</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c16">DG</font>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E2:105_1_2_7_1_3_3"><i><font color="Green" title="E151">A160</font></i></a>, <a class="ref" href="measure1.html#T62" title="MEASURE1:th.62">MEASURE1:62</a></i>;<br/><a NAME="E35:105_1_2_7_1_3_3"/><b>then </b>
<font color="Maroon" title="c3">M</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c27">B</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a> 
 <b>by </b><i><a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a>, <a class="txt" href="mesfunc5.html#E31:105_1_2_7_1_3_3"><i><font color="Green" title="E168">A177</font></i></a></i>;<br/><a NAME="E36:105_1_2_7_1_3_3"/><b>then </b>
<font color="Maroon" title="c18">e</font> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c27">B</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c18">e</font> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a> 
 <b>by </b><i><a class="txt" href="mesfunc5.html#E1:105_1_2_7_1_3"><i><font color="Green" title="E134">A142</font></i></a>, <a class="txt" href="mesfunc5.html#E2:105_1_2_7_1_3"><i><font color="Green" title="E135">A143</font></i></a>, <a class="ref" href="extreal1.html#T44" title="EXTREAL1:th.44">EXTREAL1:44</a></i>;<br/><a NAME="E37:105_1_2_7_1_3_3"/><b>then </b><i><font color="Green" title="E172">A181</font></i>: 
(  <a href="mesfunc5.html#K5" title="MESFUNC5:func.5">integral'</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c28">EP</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>  &amp;  <a href="mesfunc5.html#K5" title="MESFUNC5:func.5">integral'</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c28">EP</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a>  )
 <b>by </b><i><a class="txt" href="mesfunc5.html#E30:105_1_2_7_1_3_3"><i><font color="Green" title="E167">A176</font></i></a>, <a class="txt" href="mesfunc5.html#E33:105_1_2_7_1_3_3"><i><font color="Green" title="E170">A179</font></i></a>, <a class="txt" href="mesfunc5.html#E1:105_1_2_7_1_3"><i><font color="Green" title="E134">A142</font></i></a>, <a class="ref" href="extreal2.html#T45" title="EXTREAL2:th.45">EXTREAL2:45</a>, <a class="ref" href="xxreal_0.html#T4" title="XXREAL_0:th.4">XXREAL_0:4</a></i>;<br/><a NAME="E38:105_1_2_7_1_3_3"/><i><font color="Green" title="E173">A182</font></i>: 
( <font color="Maroon" title="c4">g</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c27">B</font> <a href="mesfunc2.html#R1" title="MESFUNC2:pred.1">is_simple_func_in</a> <font color="Maroon" title="c2">S</font> &amp; <font color="Maroon" title="c4">g</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c26">A</font> <a href="mesfunc2.html#R1" title="MESFUNC2:pred.1">is_simple_func_in</a> <font color="Maroon" title="c2">S</font> )
 <b>by </b><i><a class="txt" href="mesfunc5.html#E1:105"><i><font color="Green" title="E99">A1</font></i></a>, <a class="ref" href="mesfunc5.html#T40" target="_self" title="MESFUNC5:th.40">Th40</a></i>;<br/><a NAME="E39:105_1_2_7_1_3_3"/><i><font color="Green" title="E174">A183</font></i>: 
<font color="Maroon" title="c26">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">g</font>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E4:105_1_2_7_1_3_3"><i><font color="Green" title="E152">A161</font></i></a>, <a class="ref" href="xboole_1.html#T7" title="XBOOLE_1:th.7">XBOOLE_1:7</a></i>;<br/><a NAME="E40:105_1_2_7_1_3_3"/><b>then </b><i><font color="Green" title="E175">A184</font></i>: 
(  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c26">A</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c26">A</font> &amp;  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c27">B</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c27">B</font> )
 <b>by </b><i><a class="txt" href="mesfunc5.html#E2:105_1_2_7_1_3_3"><i><font color="Green" title="E151">A160</font></i></a>, <a class="ref" href="relat_1.html#T91" title="RELAT_1:th.91">RELAT_1:91</a></i>;<br/><a NAME="E41:105_1_2_7_1_3_3"/><i><font color="Green" title="E176">A185</font></i>: 
( <font color="Maroon" title="c4">g</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c26">A</font> is <a href="supinf_2.html#V6" title="SUPINF_2:attr.6">nonnegative</a> &amp; <font color="Maroon" title="c4">g</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c27">B</font> is <a href="supinf_2.html#V6" title="SUPINF_2:attr.6">nonnegative</a> )
 <b>by </b><i><a class="txt" href="mesfunc5.html#E9:105_1_2_7_1"><i><font color="Green" title="E129">A131</font></i></a>, <a class="ref" href="mesfunc5.html#T21" target="_self" title="MESFUNC5:th.21">Th21</a></i>;<br/><a NAME="E42:105_1_2_7_1_3_3"/><b>then </b><i><font color="Green" title="E177">A186</font></i>: 
 <a href="mesfunc5.html#K5" title="MESFUNC5:func.5">integral'</a> <font color="Maroon" title="c3">M</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c26">A</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a> 
 <b>by </b><i><a class="ref" href="mesfunc5.html#T74" target="_self" title="MESFUNC5:th.74">Th74</a>, <a class="txt" href="mesfunc5.html#E38:105_1_2_7_1_3_3"><i><font color="Green" title="E173">A182</font></i></a></i>;<br/><a NAME="E43:105_1_2_7_1_3_3"/><i><font color="Green" title="E178">A187</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c25">n</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c27">B</font></span>)</span> <a href="mesfunc1.html#K13" title="MESFUNC1:func.13">+</a> <font color="Maroon" title="c28">EP</font> <a href="mesfunc2.html#R1" title="MESFUNC2:pred.1">is_simple_func_in</a> <font color="Maroon" title="c2">S</font>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E23:105_1_2_7_1_3_3"><i><font color="Green" title="E163">A172</font></i></a>, <a class="txt" href="mesfunc5.html#E29:105_1_2_7_1_3_3"><i><font color="Green" title="E166">A175</font></i></a>, <a class="ref" href="mesfunc5.html#T44" target="_self" title="MESFUNC5:th.44">Th44</a></i>;<br/><a NAME="E44:105_1_2_7_1_3_3"/>
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c28">EP</font> holds <br/>0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c28">EP</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Olive" title="b1">x</font>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E29:105_1_2_7_1_3_3"><i><font color="Green" title="E166">A175</font></i></a>, <a class="txt" href="mesfunc5.html#E1:105_1_2_7_1_3"><i><font color="Green" title="E134">A142</font></i></a></i>;<br/><a NAME="E45:105_1_2_7_1_3_3"/><b>then </b><i><font color="Green" title="E179">A188</font></i>: 
<font color="Maroon" title="c28">EP</font> is <a href="supinf_2.html#V6" title="SUPINF_2:attr.6">nonnegative</a>
 <b>by </b><i><a class="ref" href="supinf_2.html#T71" title="SUPINF_2:th.71">SUPINF_2:71</a></i>;<br/><b>then </b><i><font color="Green" title="E180">A189</font></i>:  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c25">n</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c27">B</font></span>)</span> <a href="mesfunc1.html#K13" title="MESFUNC1:func.13">+</a> <font color="Maroon" title="c28">EP</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c25">n</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c27">B</font></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c28">EP</font></span>)</span>
<b>by </b><i><a class="txt" href="mesfunc5.html#E23:105_1_2_7_1_3_3"><i><font color="Green" title="E163">A172</font></i></a>, <a class="txt" href="mesfunc5.html#E29:105_1_2_7_1_3_3"><i><font color="Green" title="E166">A175</font></i></a>, <a class="txt" href="mesfunc5.html#E19:105_1_2_7_1_3_3"><i><font color="Green" title="E159">A168</font></i></a>, <a class="ref" href="mesfunc5.html#T71" target="_self" title="MESFUNC5:th.71">Th71</a></i>
<br/>.= 
<font color="Maroon" title="c27">B</font>
<b>by </b><i><a class="txt" href="mesfunc5.html#E29:105_1_2_7_1_3_3"><i><font color="Green" title="E166">A175</font></i></a>, <a class="txt" href="mesfunc5.html#E27:105_1_2_7_1_3_3"><i><font color="Green" title="E165">A174</font></i></a></i>
;<br/><a NAME="E47:105_1_2_7_1_3_3"/><i><font color="Green" title="E181">A190</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c25">n</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c27">B</font></span>)</span> <a href="mesfunc1.html#K13" title="MESFUNC1:func.13">+</a> <font color="Maroon" title="c28">EP</font> is <a href="supinf_2.html#V6" title="SUPINF_2:attr.6">nonnegative</a>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E19:105_1_2_7_1_3_3"><i><font color="Green" title="E159">A168</font></i></a>, <a class="txt" href="mesfunc5.html#E45:105_1_2_7_1_3_3"><i><font color="Green" title="E179">A188</font></i></a>, <a class="ref" href="mesfunc5.html#T25" target="_self" title="MESFUNC5:th.25">Th25</a></i>;<br/><a NAME="E48:105_1_2_7_1_3_3"/><i><font color="Green" title="E182">A191</font></i>: 
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c25">n</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c27">B</font></span>)</span> <a href="mesfunc1.html#K13" title="MESFUNC1:func.13">+</a> <font color="Maroon" title="c28">EP</font></span>)</span> <a href="mesfunc1.html#K6" title="MESFUNC1:func.6">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c27">B</font></span>)</span></span>)</span> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c27">B</font></span>)</span> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Olive" title="b1">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c25">n</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c27">B</font></span>)</span> <a href="mesfunc1.html#K13" title="MESFUNC1:func.13">+</a> <font color="Maroon" title="c28">EP</font></span>)</span> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Olive" title="b1">x</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="105_1_2_7_1_3_3_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c29">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:105_1_2_7_1_3_3_2"/>
<font color="Maroon" title="c29">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c25">n</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c27">B</font></span>)</span> <a href="mesfunc1.html#K13" title="MESFUNC1:func.13">+</a> <font color="Maroon" title="c28">EP</font></span>)</span> <a href="mesfunc1.html#K6" title="MESFUNC1:func.6">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c27">B</font></span>)</span></span>)</span>
 ;<br/>

<a NAME="E2:105_1_2_7_1_3_3_2"/><b>then </b>
<font color="Maroon" title="c29">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c25">n</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c27">B</font></span>)</span> <a href="mesfunc1.html#K13" title="MESFUNC1:func.13">+</a> <font color="Maroon" title="c28">EP</font></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c27">B</font></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c25">n</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c27">B</font></span>)</span> <a href="mesfunc1.html#K13" title="MESFUNC1:func.13">+</a> <font color="Maroon" title="c28">EP</font></span>)</span> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p4"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c27">B</font></span>)</span> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p4"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c25">n</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c27">B</font></span>)</span> <a href="mesfunc1.html#K13" title="MESFUNC1:func.13">+</a> <font color="Maroon" title="c28">EP</font></span>)</span> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p4"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c27">B</font></span>)</span> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p4"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="mesfunc1.html#D4" title="MESFUNC1:def.4">MESFUNC1:def 4</a></i>;<br/>
<a NAME="E3:105_1_2_7_1_3_3_2"/><b>then </b><i><font color="Green" title="E183">A192</font></i>: 
<font color="Maroon" title="c29">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c25">n</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c27">B</font></span>)</span> <a href="mesfunc1.html#K13" title="MESFUNC1:func.13">+</a> <font color="Maroon" title="c28">EP</font></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c27">B</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/>
<a NAME="E4:105_1_2_7_1_3_3_2"/><b>then </b><i><font color="Green" title="E184">A193</font></i>: 
<font color="Maroon" title="c29">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c25">n</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c27">B</font></span>)</span> <a href="mesfunc1.html#K13" title="MESFUNC1:func.13">+</a> <font color="Maroon" title="c28">EP</font></span>)</span>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E5:105_1_2_7_1_3_3_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c25">n</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c27">B</font></span>)</span> <a href="mesfunc1.html#K13" title="MESFUNC1:func.13">+</a> <font color="Maroon" title="c28">EP</font></span>)</span> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c29">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c25">n</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c27">B</font></span>)</span> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c29">x</font></span>)</span> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c28">EP</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c29">x</font></span>)</span>
 
<b>by </b><i><a class="ref" href="mesfunc1.html#D3" title="MESFUNC1:def.3">MESFUNC1:def 3</a></i>;<br/>
<a NAME="E6:105_1_2_7_1_3_3_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c25">n</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c27">B</font></span>)</span> <a href="mesfunc1.html#K13" title="MESFUNC1:func.13">+</a> <font color="Maroon" title="c28">EP</font></span>)</span> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c29">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c25">n</font></span>)</span> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c29">x</font></span>)</span> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c28">EP</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c29">x</font></span>)</span>
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E4:105_1_2_7_1_3_3_2"><i><font color="Green" title="E184">A193</font></i></a>, <a class="txt" href="mesfunc5.html#E27:105_1_2_7_1_3_3"><i><font color="Green" title="E165">A174</font></i></a>, <a class="txt" href="mesfunc5.html#E46:105_1_2_7_1_3_3"><i><font color="Green" title="E180">A189</font></i></a>, <a class="ref" href="funct_1.html#T68" title="FUNCT_1:th.68">FUNCT_1:68</a></i>;<br/>
<a NAME="E7:105_1_2_7_1_3_3_2"/><b>then </b><i><font color="Green" title="E185">A194</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c25">n</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c27">B</font></span>)</span> <a href="mesfunc1.html#K13" title="MESFUNC1:func.13">+</a> <font color="Maroon" title="c28">EP</font></span>)</span> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c29">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c25">n</font></span>)</span> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c29">x</font></span>)</span> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <font color="Maroon" title="c18">e</font>
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E4:105_1_2_7_1_3_3_2"><i><font color="Green" title="E184">A193</font></i></a>, <a class="txt" href="mesfunc5.html#E29:105_1_2_7_1_3_3"><i><font color="Green" title="E166">A175</font></i></a>, <a class="txt" href="mesfunc5.html#E46:105_1_2_7_1_3_3"><i><font color="Green" title="E180">A189</font></i></a></i>;<br/>
<a NAME="E8:105_1_2_7_1_3_3_2"/><i><font color="Green" title="E186">A195</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c27">B</font></span>)</span> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c29">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">g</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c29">x</font>
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E3:105_1_2_7_1_3_3_2"><i><font color="Green" title="E183">A192</font></i></a>, <a class="txt" href="mesfunc5.html#E46:105_1_2_7_1_3_3"><i><font color="Green" title="E180">A189</font></i></a>, <a class="txt" href="mesfunc5.html#E40:105_1_2_7_1_3_3"><i><font color="Green" title="E175">A184</font></i></a>, <a class="ref" href="funct_1.html#T68" title="FUNCT_1:th.68">FUNCT_1:68</a></i>;<br/>
<b>set </b><font color="Maroon" title="c30">f</font> = <font color="Maroon" title="c4">g</font> <a href="mesfunc1.html#K6" title="MESFUNC1:func.6">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c25">n</font></span>)</span>;<br/>
<a NAME="E9:105_1_2_7_1_3_3_2"/><i><font color="Green" title="E187">A196</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="mesfunc1.html#K6" title="MESFUNC1:func.6">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c25">n</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">g</font>
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E30:105_1_2"><i><font color="Green" title="E120">A34</font></i></a></i>;<br/>
<a NAME="E10:105_1_2_7_1_3_3_2"/>
<font color="Maroon" title="c29">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="mesfunc1.html#K15" title="MESFUNC1:func.15">less_dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="mesfunc1.html#K6" title="MESFUNC1:func.6">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c25">n</font></span>)</span></span>)</span>,<font color="Maroon" title="c18">e</font>
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E4:105_1_2_7_1_3_3_2"><i><font color="Green" title="E184">A193</font></i></a>, <a class="txt" href="mesfunc5.html#E46:105_1_2_7_1_3_3"><i><font color="Green" title="E180">A189</font></i></a>, <a class="txt" href="mesfunc5.html#E5:105_1_2_7_1_3"><i><font color="Green" title="E137">A145</font></i></a></i>;<br/>
<a NAME="E11:105_1_2_7_1_3_3_2"/><b>then </b><i><font color="Green" title="E188">A197</font></i>: 
( <font color="Maroon" title="c29">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="mesfunc1.html#K6" title="MESFUNC1:func.6">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c25">n</font></span>)</span></span>)</span> &amp;  ex <font color="Olive" title="b1">y</font> being  <a href="supinf_1.html#NM3" title="SUPINF_1:NM.3">R_eal</a> st <br/>( <font color="Olive" title="b1">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="mesfunc1.html#K6" title="MESFUNC1:func.6">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c25">n</font></span>)</span></span>)</span> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c29">x</font> &amp; <font color="Olive" title="b1">y</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c18">e</font> ) )
 
<b>by </b><i><a class="ref" href="mesfunc1.html#D12" title="MESFUNC1:def.12">MESFUNC1:def 12</a></i>;<br/>
<a NAME="E12:105_1_2_7_1_3_3_2"/><b>then </b><i><font color="Green" title="E189">A198</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c29">x</font></span>)</span> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c25">n</font></span>)</span> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c29">x</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c18">e</font>
 
<b>by </b><i><a class="ref" href="mesfunc1.html#D4" title="MESFUNC1:def.4">MESFUNC1:def 4</a></i>;<br/>
<a NAME="E13:105_1_2_7_1_3_3_2"/>
( <font color="Maroon" title="c4">g</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c29">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a>  &amp; <font color="Maroon" title="c4">g</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c29">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>  )
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E11:105_1_2_7_1_3_3_2"><i><font color="Green" title="E188">A197</font></i></a>, <a class="txt" href="mesfunc5.html#E9:105_1_2_7_1_3_3_2"><i><font color="Green" title="E187">A196</font></i></a>, <a class="txt" href="mesfunc5.html#E3:105_1_2_7_1_3"><i><font color="Green" title="E136">A144</font></i></a></i>;<br/>
<b>hence </b><a NAME="E14:105_1_2_7_1_3_3_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c27">B</font></span>)</span> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c29">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c25">n</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c27">B</font></span>)</span> <a href="mesfunc1.html#K13" title="MESFUNC1:func.13">+</a> <font color="Maroon" title="c28">EP</font></span>)</span> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c29">x</font>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E8:105_1_2_7_1_3_3_2"><i><font color="Green" title="E186">A195</font></i></a>, <a class="txt" href="mesfunc5.html#E7:105_1_2_7_1_3_3_2"><i><font color="Green" title="E185">A194</font></i></a>, <a class="txt" href="mesfunc5.html#E12:105_1_2_7_1_3_3_2"><i><font color="Green" title="E189">A198</font></i></a>, <a class="ref" href="extreal2.html#T33" title="EXTREAL2:th.33">EXTREAL2:33</a>, <a class="txt" href="mesfunc5.html#E2:105_1_2_7_1_3"><i><font color="Green" title="E135">A143</font></i></a></i>;<br/>


</div><b>end;</b></div><a NAME="E49:105_1_2_7_1_3_3"/><b>then </b><i><font color="Green" title="E183">A199</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c25">n</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c27">B</font></span>)</span> <a href="mesfunc1.html#K13" title="MESFUNC1:func.13">+</a> <font color="Maroon" title="c28">EP</font></span>)</span> <a href="mesfunc1.html#K6" title="MESFUNC1:func.6">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c27">B</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c25">n</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c27">B</font></span>)</span> <a href="mesfunc1.html#K13" title="MESFUNC1:func.13">+</a> <font color="Maroon" title="c28">EP</font></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c27">B</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="mesfunc5.html#T75" target="_self" title="MESFUNC5:th.75">Th75</a>, <a class="txt" href="mesfunc5.html#E43:105_1_2_7_1_3_3"><i><font color="Green" title="E178">A187</font></i></a>, <a class="txt" href="mesfunc5.html#E38:105_1_2_7_1_3_3"><i><font color="Green" title="E173">A182</font></i></a>, <a class="txt" href="mesfunc5.html#E47:105_1_2_7_1_3_3"><i><font color="Green" title="E181">A190</font></i></a>, <a class="txt" href="mesfunc5.html#E41:105_1_2_7_1_3_3"><i><font color="Green" title="E176">A185</font></i></a></i>;<br/><a NAME="E50:105_1_2_7_1_3_3"/>
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c25">n</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c27">B</font></span>)</span> <a href="mesfunc1.html#K13" title="MESFUNC1:func.13">+</a> <font color="Maroon" title="c28">EP</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c25">n</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c27">B</font></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c28">EP</font></span>)</span>
 <b>by </b><i><a class="ref" href="mesfunc5.html#T71" target="_self" title="MESFUNC5:th.71">Th71</a>, <a class="txt" href="mesfunc5.html#E23:105_1_2_7_1_3_3"><i><font color="Green" title="E163">A172</font></i></a>, <a class="txt" href="mesfunc5.html#E29:105_1_2_7_1_3_3"><i><font color="Green" title="E166">A175</font></i></a>, <a class="txt" href="mesfunc5.html#E19:105_1_2_7_1_3_3"><i><font color="Green" title="E159">A168</font></i></a>, <a class="txt" href="mesfunc5.html#E45:105_1_2_7_1_3_3"><i><font color="Green" title="E179">A188</font></i></a></i>;<br/><a NAME="E51:105_1_2_7_1_3_3"/><b>then </b>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c25">n</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c27">B</font></span>)</span> <a href="mesfunc1.html#K13" title="MESFUNC1:func.13">+</a> <font color="Maroon" title="c28">EP</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c25">n</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c27">B</font></span>)</span> <a href="mesfunc1.html#K13" title="MESFUNC1:func.13">+</a> <font color="Maroon" title="c28">EP</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c25">n</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c27">B</font></span>)</span> <a href="mesfunc1.html#K13" title="MESFUNC1:func.13">+</a> <font color="Maroon" title="c28">EP</font></span>)</span> <a href="mesfunc1.html#K6" title="MESFUNC1:func.6">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c27">B</font></span>)</span></span>)</span></span>)</span> &amp; <font color="Maroon" title="c4">g</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c27">B</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c27">B</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c25">n</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c27">B</font></span>)</span> <a href="mesfunc1.html#K13" title="MESFUNC1:func.13">+</a> <font color="Maroon" title="c28">EP</font></span>)</span> <a href="mesfunc1.html#K6" title="MESFUNC1:func.6">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c27">B</font></span>)</span></span>)</span></span>)</span> )
 <b>by </b><i><a class="ref" href="grfunc_1.html#T64" title="GRFUNC_1:th.64">GRFUNC_1:64</a>, <a class="txt" href="mesfunc5.html#E49:105_1_2_7_1_3_3"><i><font color="Green" title="E183">A199</font></i></a>, <a class="txt" href="mesfunc5.html#E40:105_1_2_7_1_3_3"><i><font color="Green" title="E175">A184</font></i></a>, <a class="txt" href="mesfunc5.html#E27:105_1_2_7_1_3_3"><i><font color="Green" title="E165">A174</font></i></a>, <a class="txt" href="mesfunc5.html#E29:105_1_2_7_1_3_3"><i><font color="Green" title="E166">A175</font></i></a></i>;<br/><a NAME="E52:105_1_2_7_1_3_3"/><b>then </b><i><font color="Green" title="E184">A200</font></i>: 
 <a href="mesfunc5.html#K5" title="MESFUNC5:func.5">integral'</a> <font color="Maroon" title="c3">M</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c27">B</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="mesfunc5.html#K5" title="MESFUNC5:func.5">integral'</a> <font color="Maroon" title="c3">M</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c25">n</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c27">B</font></span>)</span> <a href="mesfunc1.html#K13" title="MESFUNC1:func.13">+</a> <font color="Maroon" title="c28">EP</font></span>)</span>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E48:105_1_2_7_1_3_3"><i><font color="Green" title="E182">A191</font></i></a>, <a class="ref" href="mesfunc5.html#T76" target="_self" title="MESFUNC5:th.76">Th76</a>, <a class="txt" href="mesfunc5.html#E43:105_1_2_7_1_3_3"><i><font color="Green" title="E178">A187</font></i></a>, <a class="txt" href="mesfunc5.html#E38:105_1_2_7_1_3_3"><i><font color="Green" title="E173">A182</font></i></a>, <a class="txt" href="mesfunc5.html#E47:105_1_2_7_1_3_3"><i><font color="Green" title="E181">A190</font></i></a>, <a class="txt" href="mesfunc5.html#E41:105_1_2_7_1_3_3"><i><font color="Green" title="E176">A185</font></i></a></i>;<br/> <a href="mesfunc5.html#K5" title="MESFUNC5:func.5">integral'</a> <font color="Maroon" title="c3">M</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c25">n</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c27">B</font></span>)</span> <a href="mesfunc1.html#K13" title="MESFUNC1:func.13">+</a> <font color="Maroon" title="c28">EP</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="mesfunc5.html#K5" title="MESFUNC5:func.5">integral'</a> <font color="Maroon" title="c3">M</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c25">n</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c27">B</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c27">B</font></span>)</span></span>)</span> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <span class="p1">(<span class="default"><a href="mesfunc5.html#K5" title="MESFUNC5:func.5">integral'</a> <font color="Maroon" title="c3">M</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c28">EP</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c27">B</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="mesfunc5.html#E46:105_1_2_7_1_3_3"><i><font color="Green" title="E180">A189</font></i></a>, <a class="txt" href="mesfunc5.html#E29:105_1_2_7_1_3_3"><i><font color="Green" title="E166">A175</font></i></a>, <a class="txt" href="mesfunc5.html#E19:105_1_2_7_1_3_3"><i><font color="Green" title="E159">A168</font></i></a>, <a class="txt" href="mesfunc5.html#E23:105_1_2_7_1_3_3"><i><font color="Green" title="E163">A172</font></i></a>, <a class="txt" href="mesfunc5.html#E45:105_1_2_7_1_3_3"><i><font color="Green" title="E179">A188</font></i></a>, <a class="ref" href="mesfunc5.html#T71" target="_self" title="MESFUNC5:th.71">Th71</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="mesfunc5.html#K5" title="MESFUNC5:func.5">integral'</a> <font color="Maroon" title="c3">M</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c25">n</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c27">B</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c27">B</font></span>)</span></span>)</span> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <span class="p1">(<span class="default"><a href="mesfunc5.html#K5" title="MESFUNC5:func.5">integral'</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c28">EP</font></span>)</span>
<b>by </b><i><a class="txt" href="mesfunc5.html#E29:105_1_2_7_1_3_3"><i><font color="Green" title="E166">A175</font></i></a>, <a class="ref" href="grfunc_1.html#T64" title="GRFUNC_1:th.64">GRFUNC_1:64</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="mesfunc5.html#K5" title="MESFUNC5:func.5">integral'</a> <font color="Maroon" title="c3">M</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c25">n</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c27">B</font></span>)</span></span>)</span> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <span class="p1">(<span class="default"><a href="mesfunc5.html#K5" title="MESFUNC5:func.5">integral'</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c28">EP</font></span>)</span>
<b>by </b><i><a class="ref" href="funct_1.html#T82" title="FUNCT_1:th.82">FUNCT_1:82</a></i>
;<br/><a NAME="E54:105_1_2_7_1_3_3"/><b>then </b><i><font color="Green" title="E185">A201</font></i>: 
<span class="p1">(<span class="default"><a href="mesfunc5.html#K5" title="MESFUNC5:func.5">integral'</a> <font color="Maroon" title="c3">M</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c27">B</font></span>)</span></span>)</span> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <span class="p1">(<span class="default"><a href="mesfunc5.html#K5" title="MESFUNC5:func.5">integral'</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c28">EP</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="mesfunc5.html#K5" title="MESFUNC5:func.5">integral'</a> <font color="Maroon" title="c3">M</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c25">n</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c27">B</font></span>)</span>
 <b>by </b><i><a class="ref" href="extreal2.html#T34" title="EXTREAL2:th.34">EXTREAL2:34</a>, <a class="txt" href="mesfunc5.html#E37:105_1_2_7_1_3_3"><i><font color="Green" title="E172">A181</font></i></a>, <a class="txt" href="mesfunc5.html#E52:105_1_2_7_1_3_3"><i><font color="Green" title="E184">A200</font></i></a></i>;<br/><a NAME="E55:105_1_2_7_1_3_3"/><i><font color="Green" title="E186">A202</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c26">A</font> <a href="measure1.html#K2" title="MEASURE1:func.2">\/</a> <font color="Maroon" title="c27">B</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">g</font></span>)</span>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E4:105_1_2_7_1_3_3"><i><font color="Green" title="E152">A161</font></i></a></i>;<br/><a NAME="E56:105_1_2_7_1_3_3"/>
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">g</font> holds <br/><font color="Maroon" title="c4">g</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">g</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Olive" title="b1">x</font>
 ;<br/><a NAME="E57:105_1_2_7_1_3_3"/><b>then </b>
<font color="Maroon" title="c4">g</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c4">g</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c26">A</font> <a href="measure1.html#K2" title="MEASURE1:func.2">\/</a> <font color="Maroon" title="c27">B</font></span>)</span>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E55:105_1_2_7_1_3_3"><i><font color="Green" title="E186">A202</font></i></a>, <a class="ref" href="funct_1.html#T68" title="FUNCT_1:th.68">FUNCT_1:68</a></i>;<br/><a NAME="E58:105_1_2_7_1_3_3"/><b>then </b><i><font color="Green" title="E187">A203</font></i>: 
 <a href="mesfunc5.html#K5" title="MESFUNC5:func.5">integral'</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="mesfunc5.html#K5" title="MESFUNC5:func.5">integral'</a> <font color="Maroon" title="c3">M</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c26">A</font></span>)</span></span>)</span> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <span class="p1">(<span class="default"><a href="mesfunc5.html#K5" title="MESFUNC5:func.5">integral'</a> <font color="Maroon" title="c3">M</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c27">B</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E20:105_1_2_7_1_3_3"><i><font color="Green" title="E160">A169</font></i></a>, <a class="ref" href="mesfunc5.html#T73" target="_self" title="MESFUNC5:th.73">Th73</a>, <a class="txt" href="mesfunc5.html#E1:105"><i><font color="Green" title="E99">A1</font></i></a>, <a class="txt" href="mesfunc5.html#E9:105_1_2_7_1"><i><font color="Green" title="E129">A131</font></i></a></i>;<br/><b>consider </b><font color="Maroon" title="c29">GP</font> being   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <font color="Maroon" title="c1">X</font>, <a href="numbers.html#K6" title="NUMBERS:func.6">ExtREAL</a> <b> such that </b><br/><a NAME="E60:105_1_2_7_1_3_3"/><i><font color="Green" title="E188">A204</font></i>: 
( <font color="Maroon" title="c29">GP</font> <a href="mesfunc2.html#R1" title="MESFUNC2:pred.1">is_simple_func_in</a> <font color="Maroon" title="c2">S</font> &amp;  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c29">GP</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c26">A</font> &amp; ( for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c26">A</font> holds <br/><font color="Maroon" title="c29">GP</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <span class="p1">(<span class="default"><a href="pre_circ.html#K1" title="PRE_CIRC:func.1">max</a> <font color="Maroon" title="c10">RINGA</font></span>)</span> ) )
 <b>by </b><i><a class="ref" href="mesfunc5.html#T47" target="_self" title="MESFUNC5:th.47">Th47</a></i>;<br/><a NAME="E61:105_1_2_7_1_3_3"/>
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c29">GP</font> holds <br/>0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c29">GP</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Olive" title="b1">x</font>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E60:105_1_2_7_1_3_3"><i><font color="Green" title="E188">A204</font></i></a>, <a class="txt" href="mesfunc5.html#E10:105_1_2_7_1"><i><font color="Green" title="E130">A132</font></i></a></i>;<br/><a NAME="E62:105_1_2_7_1_3_3"/><b>then </b><i><font color="Green" title="E189">A205</font></i>: 
<font color="Maroon" title="c29">GP</font> is <a href="supinf_2.html#V6" title="SUPINF_2:attr.6">nonnegative</a>
 <b>by </b><i><a class="ref" href="supinf_2.html#T71" title="SUPINF_2:th.71">SUPINF_2:71</a></i>;<br/><a NAME="E63:105_1_2_7_1_3_3"/><i><font color="Green" title="E190">A206</font></i>: 
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c29">GP</font> <a href="mesfunc1.html#K6" title="MESFUNC1:func.6">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c26">A</font></span>)</span></span>)</span> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c26">A</font></span>)</span> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Olive" title="b1">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c29">GP</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Olive" title="b1">x</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="105_1_2_7_1_3_3_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c30">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:105_1_2_7_1_3_3_4"/>
<font color="Maroon" title="c30">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c29">GP</font> <a href="mesfunc1.html#K6" title="MESFUNC1:func.6">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c26">A</font></span>)</span></span>)</span>
 ;<br/>

<a NAME="E2:105_1_2_7_1_3_3_4"/><b>then </b>
<font color="Maroon" title="c30">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c29">GP</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c26">A</font></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c29">GP</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p4"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c26">A</font></span>)</span> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p4"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c29">GP</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p4"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c26">A</font></span>)</span> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p4"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="mesfunc1.html#D4" title="MESFUNC1:def.4">MESFUNC1:def 4</a></i>;<br/>
<a NAME="E3:105_1_2_7_1_3_3_4"/><b>then </b><i><font color="Green" title="E191">A207</font></i>: 
<font color="Maroon" title="c30">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c29">GP</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c26">A</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/>
<a NAME="E4:105_1_2_7_1_3_3_4"/><b>then </b><i><font color="Green" title="E192">A208</font></i>: 
<font color="Maroon" title="c30">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c29">GP</font>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E5:105_1_2_7_1_3_3_4"/><i><font color="Green" title="E193">A209</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c26">A</font></span>)</span> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c30">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">g</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c30">x</font>
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E40:105_1_2_7_1_3_3"><i><font color="Green" title="E175">A184</font></i></a>, <a class="txt" href="mesfunc5.html#E60:105_1_2_7_1_3_3"><i><font color="Green" title="E188">A204</font></i></a>, <a class="txt" href="mesfunc5.html#E3:105_1_2_7_1_3_3_4"><i><font color="Green" title="E191">A207</font></i></a>, <a class="ref" href="funct_1.html#T70" title="FUNCT_1:th.70">FUNCT_1:70</a></i>;<br/>
<a NAME="E6:105_1_2_7_1_3_3_4"/>
<font color="Maroon" title="c30">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">g</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c26">A</font>
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E40:105_1_2_7_1_3_3"><i><font color="Green" title="E175">A184</font></i></a>, <a class="txt" href="mesfunc5.html#E60:105_1_2_7_1_3_3"><i><font color="Green" title="E188">A204</font></i></a>, <a class="txt" href="mesfunc5.html#E3:105_1_2_7_1_3_3_4"><i><font color="Green" title="E191">A207</font></i></a>, <a class="ref" href="funct_1.html#T68" title="FUNCT_1:th.68">FUNCT_1:68</a></i>;<br/>
<a NAME="E7:105_1_2_7_1_3_3_4"/><b>then </b>
<font color="Maroon" title="c30">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">g</font>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E8:105_1_2_7_1_3_3_4"/><b>then </b>
<font color="Maroon" title="c4">g</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c30">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <span class="p1">(<span class="default"><a href="pre_circ.html#K1" title="PRE_CIRC:func.1">max</a> <font color="Maroon" title="c10">RINGA</font></span>)</span>
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E34:105_1_2"><i><font color="Green" title="E124">A117</font></i></a></i>;<br/>
<b>hence </b><a NAME="E9:105_1_2_7_1_3_3_4"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c26">A</font></span>)</span> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c30">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c29">GP</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c30">x</font>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E5:105_1_2_7_1_3_3_4"><i><font color="Green" title="E193">A209</font></i></a>, <a class="txt" href="mesfunc5.html#E4:105_1_2_7_1_3_3_4"><i><font color="Green" title="E192">A208</font></i></a>, <a class="txt" href="mesfunc5.html#E60:105_1_2_7_1_3_3"><i><font color="Green" title="E188">A204</font></i></a></i>;<br/>


</div><b>end;</b></div><a NAME="E64:105_1_2_7_1_3_3"/><b>then </b>
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c29">GP</font> <a href="mesfunc1.html#K6" title="MESFUNC1:func.6">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c26">A</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c29">GP</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c26">A</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="mesfunc5.html#T75" target="_self" title="MESFUNC5:th.75">Th75</a>, <a class="txt" href="mesfunc5.html#E38:105_1_2_7_1_3_3"><i><font color="Green" title="E173">A182</font></i></a>, <a class="txt" href="mesfunc5.html#E60:105_1_2_7_1_3_3"><i><font color="Green" title="E188">A204</font></i></a>, <a class="txt" href="mesfunc5.html#E41:105_1_2_7_1_3_3"><i><font color="Green" title="E176">A185</font></i></a>, <a class="txt" href="mesfunc5.html#E62:105_1_2_7_1_3_3"><i><font color="Green" title="E189">A205</font></i></a></i>;<br/><a NAME="E65:105_1_2_7_1_3_3"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c26">A</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c29">GP</font> <a href="mesfunc1.html#K6" title="MESFUNC1:func.6">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c26">A</font></span>)</span></span>)</span></span>)</span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c4">g</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c26">A</font> &amp; <font color="Maroon" title="c29">GP</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c29">GP</font> <a href="mesfunc1.html#K6" title="MESFUNC1:func.6">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c26">A</font></span>)</span></span>)</span></span>)</span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c29">GP</font> )
 <b>by </b><i><a class="ref" href="grfunc_1.html#T64" title="GRFUNC_1:th.64">GRFUNC_1:64</a>, <a class="txt" href="mesfunc5.html#E60:105_1_2_7_1_3_3"><i><font color="Green" title="E188">A204</font></i></a>, <a class="txt" href="mesfunc5.html#E40:105_1_2_7_1_3_3"><i><font color="Green" title="E175">A184</font></i></a></i>;<br/><a NAME="E66:105_1_2_7_1_3_3"/><b>then </b><i><font color="Green" title="E191">A210</font></i>: 
 <a href="mesfunc5.html#K5" title="MESFUNC5:func.5">integral'</a> <font color="Maroon" title="c3">M</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c26">A</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="mesfunc5.html#K5" title="MESFUNC5:func.5">integral'</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c29">GP</font>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E63:105_1_2_7_1_3_3"><i><font color="Green" title="E190">A206</font></i></a>, <a class="ref" href="mesfunc5.html#T76" target="_self" title="MESFUNC5:th.76">Th76</a>, <a class="txt" href="mesfunc5.html#E38:105_1_2_7_1_3_3"><i><font color="Green" title="E173">A182</font></i></a>, <a class="txt" href="mesfunc5.html#E60:105_1_2_7_1_3_3"><i><font color="Green" title="E188">A204</font></i></a>, <a class="txt" href="mesfunc5.html#E41:105_1_2_7_1_3_3"><i><font color="Green" title="E176">A185</font></i></a>, <a class="txt" href="mesfunc5.html#E62:105_1_2_7_1_3_3"><i><font color="Green" title="E189">A205</font></i></a></i>;<br/><a NAME="E67:105_1_2_7_1_3_3"/><i><font color="Green" title="E192">A211</font></i>: 
 <a href="mesfunc5.html#K5" title="MESFUNC5:func.5">integral'</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c29">GP</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <span class="p2">(<span class="default"><a href="pre_circ.html#K1" title="PRE_CIRC:func.1">max</a> <font color="Maroon" title="c10">RINGA</font></span>)</span></span>)</span> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c26">A</font></span>)</span>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E10:105_1_2_7_1"><i><font color="Green" title="E130">A132</font></i></a>, <a class="txt" href="mesfunc5.html#E60:105_1_2_7_1_3_3"><i><font color="Green" title="E188">A204</font></i></a>, <a class="ref" href="mesfunc5.html#T77" target="_self" title="MESFUNC5:th.77">Th77</a></i>;<br/><a NAME="E68:105_1_2_7_1_3_3"/><i><font color="Green" title="E193">A212</font></i>: 
<font color="Maroon" title="c3">M</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c26">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a> 
 <b>by </b><i><a class="txt" href="mesfunc5.html#E31:105_1_2_7_1_3_3"><i><font color="Green" title="E168">A177</font></i></a>, <a class="txt" href="mesfunc5.html#E39:105_1_2_7_1_3_3"><i><font color="Green" title="E174">A183</font></i></a>, <a class="ref" href="measure1.html#T62" title="MEASURE1:th.62">MEASURE1:62</a></i>;<br/><a NAME="E69:105_1_2_7_1_3_3"/>
<font color="Maroon" title="c3">M</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c26">A</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">M</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c16">DG</font>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E39:105_1_2_7_1_3_3"><i><font color="Green" title="E174">A183</font></i></a>, <a class="ref" href="measure1.html#T62" title="MEASURE1:th.62">MEASURE1:62</a></i>;<br/><a NAME="E70:105_1_2_7_1_3_3"/><b>then </b>
<font color="Maroon" title="c3">M</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c26">A</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a> 
 <b>by </b><i><a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a>, <a class="txt" href="mesfunc5.html#E31:105_1_2_7_1_3_3"><i><font color="Green" title="E168">A177</font></i></a></i>;<br/><a NAME="E71:105_1_2_7_1_3_3"/><b>then </b>
<span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <span class="p2">(<span class="default"><a href="pre_circ.html#K1" title="PRE_CIRC:func.1">max</a> <font color="Maroon" title="c10">RINGA</font></span>)</span></span>)</span> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c26">A</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <span class="p2">(<span class="default"><a href="pre_circ.html#K1" title="PRE_CIRC:func.1">max</a> <font color="Maroon" title="c10">RINGA</font></span>)</span></span>)</span> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a> 
 <b>by </b><i><a class="txt" href="mesfunc5.html#E10:105_1_2_7_1"><i><font color="Green" title="E130">A132</font></i></a>, <a class="ref" href="extreal1.html#T44" title="EXTREAL1:th.44">EXTREAL1:44</a></i>;<br/><a NAME="E72:105_1_2_7_1_3_3"/><b>then </b><i><font color="Green" title="E194">A213</font></i>: 
<span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <span class="p2">(<span class="default"><a href="pre_circ.html#K1" title="PRE_CIRC:func.1">max</a> <font color="Maroon" title="c10">RINGA</font></span>)</span></span>)</span> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c26">A</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a> 
 <b>by </b><i><a class="txt" href="mesfunc5.html#E10:105_1_2_7_1"><i><font color="Green" title="E130">A132</font></i></a>, <a class="ref" href="extreal1.html#D1" title="EXTREAL1:def.1">EXTREAL1:def 1</a></i>;<br/><a NAME="E73:105_1_2_7_1_3_3"/>
 <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c26">A</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T2" title="XBOOLE_1:th.2">XBOOLE_1:2</a></i>;<br/><a NAME="E74:105_1_2_7_1_3_3"/><b>then </b><i><font color="Green" title="E195">A214</font></i>: 
0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">M</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c26">A</font>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E3:105_1_2_7_1"><i><font color="Green" title="E126">A128</font></i></a>, <a class="txt" href="mesfunc5.html#E5:105_1_2_7_1"><i><font color="Green" title="E127">A129</font></i></a>, <a class="ref" href="measure1.html#T62" title="MEASURE1:th.62">MEASURE1:62</a></i>;<br/><a NAME="E75:105_1_2_7_1_3_3"/><b>then </b><i><font color="Green" title="E196">A215</font></i>: 
<span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <span class="p2">(<span class="default"><a href="pre_circ.html#K1" title="PRE_CIRC:func.1">max</a> <font color="Maroon" title="c10">RINGA</font></span>)</span></span>)</span> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c26">A</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a> 
 <b>by </b><i><a class="txt" href="mesfunc5.html#E10:105_1_2_7_1"><i><font color="Green" title="E130">A132</font></i></a>, <a class="ref" href="extreal2.html#T45" title="EXTREAL2:th.45">EXTREAL2:45</a></i>;<br/><a NAME="E76:105_1_2_7_1_3_3"/>
<font color="Maroon" title="c3">M</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">DG</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p2">(<span class="default"><font color="Maroon" title="c19">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">n</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c18">e</font>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E26:105_1_2_7_1_3"><i><font color="Green" title="E149">A157</font></i></a>, <a class="txt" href="mesfunc5.html#E1:105_1_2_7_1_3_3"><i><font color="Green" title="E150">A159</font></i></a></i>;<br/><a NAME="E77:105_1_2_7_1_3_3"/><b>then </b><i><font color="Green" title="E197">A216</font></i>: 
<span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <span class="p2">(<span class="default"><a href="pre_circ.html#K1" title="PRE_CIRC:func.1">max</a> <font color="Maroon" title="c10">RINGA</font></span>)</span></span>)</span> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c26">A</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <span class="p2">(<span class="default"><a href="pre_circ.html#K1" title="PRE_CIRC:func.1">max</a> <font color="Maroon" title="c10">RINGA</font></span>)</span></span>)</span> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <font color="Maroon" title="c18">e</font>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E10:105_1_2_7_1"><i><font color="Green" title="E130">A132</font></i></a>, <a class="ref" href="extreal1.html#T44" title="EXTREAL1:th.44">EXTREAL1:44</a></i>;<br/><a NAME="E78:105_1_2_7_1_3_3"/><b>then </b>
 <a href="mesfunc5.html#K5" title="MESFUNC5:func.5">integral'</a> <font color="Maroon" title="c3">M</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c26">A</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <span class="p2">(<span class="default"><a href="pre_circ.html#K1" title="PRE_CIRC:func.1">max</a> <font color="Maroon" title="c10">RINGA</font></span>)</span></span>)</span> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <font color="Maroon" title="c18">e</font>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E66:105_1_2_7_1_3_3"><i><font color="Green" title="E191">A210</font></i></a>, <a class="txt" href="mesfunc5.html#E67:105_1_2_7_1_3_3"><i><font color="Green" title="E192">A211</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E79:105_1_2_7_1_3_3"/><b>then </b><i><font color="Green" title="E198">A217</font></i>: 
 <a href="mesfunc5.html#K5" title="MESFUNC5:func.5">integral'</a> <font color="Maroon" title="c3">M</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c26">A</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a> 
 <b>by </b><i><a class="ref" href="xxreal_0.html#T4" title="XXREAL_0:th.4">XXREAL_0:4</a></i>;<br/><a NAME="E80:105_1_2_7_1_3_3"/>
( <font color="Maroon" title="c18">e</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>  &amp; <font color="Maroon" title="c18">e</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a>  )
 <b>by </b><i><a class="txt" href="mesfunc5.html#E1:105_1_2_7_1_3"><i><font color="Green" title="E134">A142</font></i></a>, <a class="ref" href="xxreal_0.html#T4" title="XXREAL_0:th.4">XXREAL_0:4</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c30">ee</font> = <font color="Maroon" title="c18">e</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> <b>by </b><i><a class="ref" href="xxreal_0.html#T14" title="XXREAL_0:th.14">XXREAL_0:14</a></i>;<br/><a NAME="E82:105_1_2_7_1_3_3"/><i><font color="Green" title="E199">A218</font></i>: 
<span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <span class="p2">(<span class="default"><a href="pre_circ.html#K1" title="PRE_CIRC:func.1">max</a> <font color="Maroon" title="c10">RINGA</font></span>)</span></span>)</span> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <font color="Maroon" title="c18">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">beta1</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c30">ee</font>
 <b>by </b><i><a class="ref" href="extreal1.html#T13" title="EXTREAL1:th.13">EXTREAL1:13</a></i>;<br/><a NAME="E83:105_1_2_7_1_3_3"/><i><font color="Green" title="E200">A219</font></i>: 
<font color="Maroon" title="c3">M</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c27">B</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a> 
 <b>by </b><i><a class="txt" href="mesfunc5.html#E3:105_1_2_7_1"><i><font color="Green" title="E126">A128</font></i></a>, <a class="txt" href="mesfunc5.html#E5:105_1_2_7_1"><i><font color="Green" title="E127">A129</font></i></a>, <a class="txt" href="mesfunc5.html#E32:105_1_2_7_1_3_3"><i><font color="Green" title="E169">A178</font></i></a>, <a class="ref" href="measure1.html#T62" title="MEASURE1:th.62">MEASURE1:62</a></i>;<br/><a NAME="E84:105_1_2_7_1_3_3"/>
<font color="Maroon" title="c3">M</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">g</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a> 
 <b>by </b><i><a class="txt" href="mesfunc5.html#E34:105_1_2_7_1_3_3"><i><font color="Green" title="E171">A180</font></i></a>, <a class="txt" href="mesfunc5.html#E32:105_1_2_7_1_3_3"><i><font color="Green" title="E169">A178</font></i></a>, <a class="txt" href="mesfunc5.html#E3:105_1_2_7_1"><i><font color="Green" title="E126">A128</font></i></a>, <a class="txt" href="mesfunc5.html#E5:105_1_2_7_1"><i><font color="Green" title="E127">A129</font></i></a>, <a class="ref" href="measure1.html#T62" title="MEASURE1:th.62">MEASURE1:62</a></i>;<br/><a NAME="E85:105_1_2_7_1_3_3"/><b>then </b><i><font color="Green" title="E201">A220</font></i>: 
<font color="Maroon" title="c3">M</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">g</font></span>)</span> is   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E1:105_1_2_7_1"><i><font color="Green" title="E125">A127</font></i></a>, <a class="ref" href="xxreal_0.html#T14" title="XXREAL_0:th.14">XXREAL_0:14</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c31">XSMg</font> =  <a href="mesfunc5.html#K5" title="MESFUNC5:func.5">integral'</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">g</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> <b>by </b><i><a class="txt" href="mesfunc5.html#E14:105_1_2_7_1"><i><font color="Green" title="E132">A140</font></i></a>, <a class="ref" href="xxreal_0.html#T14" title="XXREAL_0:th.14">XXREAL_0:14</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c32">XSMgA</font> =  <a href="mesfunc5.html#K5" title="MESFUNC5:func.5">integral'</a> <font color="Maroon" title="c3">M</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c26">A</font></span>)</span> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> <b>by </b><i><a class="txt" href="mesfunc5.html#E79:105_1_2_7_1_3_3"><i><font color="Green" title="E198">A217</font></i></a>, <a class="txt" href="mesfunc5.html#E42:105_1_2_7_1_3_3"><i><font color="Green" title="E177">A186</font></i></a>, <a class="ref" href="xxreal_0.html#T14" title="XXREAL_0:th.14">XXREAL_0:14</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c33">XSMGP</font> =  <a href="mesfunc5.html#K5" title="MESFUNC5:func.5">integral'</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c29">GP</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> <b>by </b><i><a class="txt" href="mesfunc5.html#E67:105_1_2_7_1_3_3"><i><font color="Green" title="E192">A211</font></i></a>, <a class="txt" href="mesfunc5.html#E72:105_1_2_7_1_3_3"><i><font color="Green" title="E194">A213</font></i></a>, <a class="txt" href="mesfunc5.html#E75:105_1_2_7_1_3_3"><i><font color="Green" title="E196">A215</font></i></a>, <a class="ref" href="xxreal_0.html#T14" title="XXREAL_0:th.14">XXREAL_0:14</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c34">MA</font> = <font color="Maroon" title="c3">M</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c26">A</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> <b>by </b><i><a class="txt" href="mesfunc5.html#E68:105_1_2_7_1_3_3"><i><font color="Green" title="E193">A212</font></i></a>, <a class="txt" href="mesfunc5.html#E74:105_1_2_7_1_3_3"><i><font color="Green" title="E195">A214</font></i></a>, <a class="ref" href="xxreal_0.html#T14" title="XXREAL_0:th.14">XXREAL_0:14</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c35">betae</font> = <span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <span class="p2">(<span class="default"><a href="pre_circ.html#K1" title="PRE_CIRC:func.1">max</a> <font color="Maroon" title="c10">RINGA</font></span>)</span></span>)</span> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <font color="Maroon" title="c18">e</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> <b>by </b><i><a class="txt" href="mesfunc5.html#E82:105_1_2_7_1_3_3"><i><font color="Green" title="E199">A218</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon" title="c36">XSMEP</font> =  <a href="mesfunc5.html#K5" title="MESFUNC5:func.5">integral'</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c28">EP</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> <b>by </b><i><a class="txt" href="mesfunc5.html#E37:105_1_2_7_1_3_3"><i><font color="Green" title="E172">A181</font></i></a>, <a class="ref" href="xxreal_0.html#T14" title="XXREAL_0:th.14">XXREAL_0:14</a></i>;<br/><a NAME="E92:105_1_2_7_1_3_3"/>
<font color="Maroon" title="c3">M</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c27">B</font> is   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E34:105_1_2_7_1_3_3"><i><font color="Green" title="E171">A180</font></i></a>, <a class="txt" href="mesfunc5.html#E83:105_1_2_7_1_3_3"><i><font color="Green" title="E200">A219</font></i></a>, <a class="txt" href="mesfunc5.html#E31:105_1_2_7_1_3_3"><i><font color="Green" title="E168">A177</font></i></a>, <a class="ref" href="xxreal_0.html#T14" title="XXREAL_0:th.14">XXREAL_0:14</a></i>;<br/><b>then consider </b><font color="Maroon" title="c37">MB</font>, <font color="Maroon" title="c38">MG</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><b> such that </b><br/><a NAME="E94:105_1_2_7_1_3_3"/><i><font color="Green" title="E202">A221</font></i>: 
( <font color="Maroon" title="c37">MB</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">M</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c27">B</font> &amp; <font color="Maroon" title="c38">MG</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">M</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">g</font></span>)</span> &amp; <font color="Maroon" title="c37">MB</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c38">MG</font> )
 <b>by </b><i><a class="txt" href="mesfunc5.html#E34:105_1_2_7_1_3_3"><i><font color="Green" title="E171">A180</font></i></a>, <a class="txt" href="mesfunc5.html#E85:105_1_2_7_1_3_3"><i><font color="Green" title="E201">A220</font></i></a></i>;<br/><i><font color="Green" title="E203">A222</font></i>: <font color="Maroon" title="c36">XSMEP</font> = 
<font color="Maroon" title="c18">e</font> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c27">B</font></span>)</span>
<b>by </b><i><a class="txt" href="mesfunc5.html#E29:105_1_2_7_1_3_3"><i><font color="Green" title="E166">A175</font></i></a>, <a class="txt" href="mesfunc5.html#E1:105_1_2_7_1_3"><i><font color="Green" title="E134">A142</font></i></a>, <a class="ref" href="mesfunc5.html#T77" target="_self" title="MESFUNC5:th.77">Th77</a></i>
<br/>.= 
<font color="Maroon" title="c30">ee</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c37">MB</font>
<b>by </b><i><a class="txt" href="mesfunc5.html#E94:105_1_2_7_1_3_3"><i><font color="Green" title="E202">A221</font></i></a>, <a class="ref" href="extreal1.html#T13" title="EXTREAL1:th.13">EXTREAL1:13</a></i>
;<br/><a NAME="E96:105_1_2_7_1_3_3"/><i><font color="Green" title="E204">A223</font></i>: 
<span class="p1">(<span class="default"><a href="mesfunc5.html#K5" title="MESFUNC5:func.5">integral'</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">g</font></span>)</span> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <span class="p1">(<span class="default"><a href="mesfunc5.html#K5" title="MESFUNC5:func.5">integral'</a> <font color="Maroon" title="c3">M</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c26">A</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mesfunc5.html#K5" title="MESFUNC5:func.5">integral'</a> <font color="Maroon" title="c3">M</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c27">B</font></span>)</span>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E79:105_1_2_7_1_3_3"><i><font color="Green" title="E198">A217</font></i></a>, <a class="txt" href="mesfunc5.html#E58:105_1_2_7_1_3_3"><i><font color="Green" title="E187">A203</font></i></a>, <a class="txt" href="mesfunc5.html#E42:105_1_2_7_1_3_3"><i><font color="Green" title="E177">A186</font></i></a>, <a class="ref" href="extreal2.html#T11" title="EXTREAL2:th.11">EXTREAL2:11</a></i>;<br/><a NAME="E97:105_1_2_7_1_3_3"/><i><font color="Green" title="E205">A224</font></i>: 
<span class="p1">(<span class="default"><a href="mesfunc5.html#K5" title="MESFUNC5:func.5">integral'</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">g</font></span>)</span> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <span class="p1">(<span class="default"><a href="mesfunc5.html#K5" title="MESFUNC5:func.5">integral'</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c29">GP</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="mesfunc5.html#K5" title="MESFUNC5:func.5">integral'</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">g</font></span>)</span> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <span class="p1">(<span class="default"><a href="mesfunc5.html#K5" title="MESFUNC5:func.5">integral'</a> <font color="Maroon" title="c3">M</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c26">A</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E79:105_1_2_7_1_3_3"><i><font color="Green" title="E198">A217</font></i></a>, <a class="txt" href="mesfunc5.html#E66:105_1_2_7_1_3_3"><i><font color="Green" title="E191">A210</font></i></a>, <a class="ref" href="supinf_2.html#T15" title="SUPINF_2:th.15">SUPINF_2:15</a>, <a class="txt" href="mesfunc5.html#E42:105_1_2_7_1_3_3"><i><font color="Green" title="E177">A186</font></i></a>, <a class="txt" href="mesfunc5.html#E67:105_1_2_7_1_3_3"><i><font color="Green" title="E192">A211</font></i></a>, <a class="txt" href="mesfunc5.html#E72:105_1_2_7_1_3_3"><i><font color="Green" title="E194">A213</font></i></a>, <a class="txt" href="mesfunc5.html#E75:105_1_2_7_1_3_3"><i><font color="Green" title="E196">A215</font></i></a></i>;<br/><font color="Maroon" title="c31">XSMg</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c32">XSMgA</font> = 
<span class="p1">(<span class="default"><a href="mesfunc5.html#K5" title="MESFUNC5:func.5">integral'</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">g</font></span>)</span> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <span class="p1">(<span class="default"><a href="mesfunc5.html#K5" title="MESFUNC5:func.5">integral'</a> <font color="Maroon" title="c3">M</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c26">A</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="supinf_2.html#T5" title="SUPINF_2:th.5">SUPINF_2:5</a></i>
<br/>.= 
 <a href="mesfunc5.html#K5" title="MESFUNC5:func.5">integral'</a> <font color="Maroon" title="c3">M</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c27">B</font></span>)</span>
<b>by </b><i><a class="txt" href="mesfunc5.html#E79:105_1_2_7_1_3_3"><i><font color="Green" title="E198">A217</font></i></a>, <a class="txt" href="mesfunc5.html#E58:105_1_2_7_1_3_3"><i><font color="Green" title="E187">A203</font></i></a>, <a class="txt" href="mesfunc5.html#E42:105_1_2_7_1_3_3"><i><font color="Green" title="E177">A186</font></i></a>, <a class="ref" href="extreal2.html#T11" title="EXTREAL2:th.11">EXTREAL2:11</a></i>
;<br/><b>then reconsider </b><font color="Maroon" title="c39">XSMgB</font> =  <a href="mesfunc5.html#K5" title="MESFUNC5:func.5">integral'</a> <font color="Maroon" title="c3">M</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c27">B</font></span>)</span> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> ;<br/><a NAME="E100:105_1_2_7_1_3_3"/><i><font color="Green" title="E206">A225</font></i>: 
<font color="Maroon" title="c31">XSMg</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c35">betae</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c31">XSMg</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c33">XSMGP</font>
 <b>by </b><i><a class="ref" href="real_1.html#T92" title="REAL_1:th.92">REAL_1:92</a>, <a class="txt" href="mesfunc5.html#E67:105_1_2_7_1_3_3"><i><font color="Green" title="E192">A211</font></i></a>, <a class="txt" href="mesfunc5.html#E77:105_1_2_7_1_3_3"><i><font color="Green" title="E197">A216</font></i></a></i>;<br/><a NAME="E101:105_1_2_7_1_3_3"/>
<span class="p1">(<span class="default"><a href="mesfunc5.html#K5" title="MESFUNC5:func.5">integral'</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">g</font></span>)</span> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <span class="p1">(<span class="default"><a href="mesfunc5.html#K5" title="MESFUNC5:func.5">integral'</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c29">GP</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c31">XSMg</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c33">XSMGP</font>
 <b>by </b><i><a class="ref" href="supinf_2.html#T5" title="SUPINF_2:th.5">SUPINF_2:5</a></i>;<br/><a NAME="E102:105_1_2_7_1_3_3"/><b>then </b>
<font color="Maroon" title="c31">XSMg</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c35">betae</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c39">XSMgB</font>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E100:105_1_2_7_1_3_3"><i><font color="Green" title="E206">A225</font></i></a>, <a class="txt" href="mesfunc5.html#E96:105_1_2_7_1_3_3"><i><font color="Green" title="E204">A223</font></i></a>, <a class="txt" href="mesfunc5.html#E97:105_1_2_7_1_3_3"><i><font color="Green" title="E205">A224</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E103:105_1_2_7_1_3_3"/><b>then </b><i><font color="Green" title="E207">A226</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c31">XSMg</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c35">betae</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c30">ee</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c37">MB</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c39">XSMgB</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c36">XSMEP</font>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E95:105_1_2_7_1_3_3"><i><font color="Green" title="E203">A222</font></i></a>, <a class="ref" href="real_1.html#T92" title="REAL_1:th.92">REAL_1:92</a></i>;<br/><a NAME="E104:105_1_2_7_1_3_3"/><i><font color="Green" title="E208">A227</font></i>: 
<font color="Maroon" title="c35">betae</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c30">ee</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c13">beta1</font>
 <b>by </b><i><a class="ref" href="extreal1.html#T13" title="EXTREAL1:th.13">EXTREAL1:13</a></i>;<br/><a NAME="E105:105_1_2_7_1_3_3"/>
<font color="Maroon" title="c30">ee</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c37">MB</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c30">ee</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c38">MG</font>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E1:105_1_2_7_1_3"><i><font color="Green" title="E134">A142</font></i></a>, <a class="ref" href="xreal_1.html#T66" title="XREAL_1:th.66">XREAL_1:66</a>, <a class="txt" href="mesfunc5.html#E94:105_1_2_7_1_3_3"><i><font color="Green" title="E202">A221</font></i></a></i>;<br/><a NAME="E106:105_1_2_7_1_3_3"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c31">XSMg</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c35">betae</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c30">ee</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c38">MG</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c31">XSMg</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c35">betae</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c30">ee</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c37">MB</font></span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T15" title="XREAL_1:th.15">XREAL_1:15</a></i>;<br/><a NAME="E107:105_1_2_7_1_3_3"/><b>then </b><i><font color="Green" title="E209">A228</font></i>: 
<font color="Maroon" title="c31">XSMg</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c30">ee</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">beta1</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <font color="Maroon" title="c38">MG</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c39">XSMgB</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c36">XSMEP</font>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E104:105_1_2_7_1_3_3"><i><font color="Green" title="E208">A227</font></i></a>, <a class="txt" href="mesfunc5.html#E103:105_1_2_7_1_3_3"><i><font color="Green" title="E207">A226</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E108:105_1_2_7_1_3_3"/>
<font color="Maroon" title="c13">beta1</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <font color="Maroon" title="c38">MG</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <span class="p2">(<span class="default"><a href="pre_circ.html#K1" title="PRE_CIRC:func.1">max</a> <font color="Maroon" title="c10">RINGA</font></span>)</span></span>)</span> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <span class="p2">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">g</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E94:105_1_2_7_1_3_3"><i><font color="Green" title="E202">A221</font></i></a>, <a class="ref" href="supinf_2.html#T1" title="SUPINF_2:th.1">SUPINF_2:1</a></i>;<br/><a NAME="E109:105_1_2_7_1_3_3"/><b>then </b>
<font color="Maroon" title="c30">ee</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">beta1</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <font color="Maroon" title="c38">MG</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">e</font> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <span class="p3">(<span class="default"><a href="pre_circ.html#K1" title="PRE_CIRC:func.1">max</a> <font color="Maroon" title="c10">RINGA</font></span>)</span></span>)</span> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <span class="p3">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">g</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="extreal1.html#T13" title="EXTREAL1:th.13">EXTREAL1:13</a></i>;<br/><a NAME="E110:105_1_2_7_1_3_3"/><b>then </b><i><font color="Green" title="E210">A229</font></i>: 
<font color="Maroon" title="c31">XSMg</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c30">ee</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">beta1</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <font color="Maroon" title="c38">MG</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="mesfunc5.html#K5" title="MESFUNC5:func.5">integral'</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">g</font></span>)</span> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">e</font> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <span class="p4">(<span class="default"><a href="pre_circ.html#K1" title="PRE_CIRC:func.1">max</a> <font color="Maroon" title="c10">RINGA</font></span>)</span></span>)</span> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <span class="p4">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">g</font></span>)</span></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="supinf_2.html#T5" title="SUPINF_2:th.5">SUPINF_2:5</a></i>;<br/><a NAME="E111:105_1_2_7_1_3_3"/>
<font color="Maroon" title="c39">XSMgB</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c36">XSMEP</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="mesfunc5.html#K5" title="MESFUNC5:func.5">integral'</a> <font color="Maroon" title="c3">M</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c27">B</font></span>)</span></span>)</span> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <span class="p1">(<span class="default"><a href="mesfunc5.html#K5" title="MESFUNC5:func.5">integral'</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c28">EP</font></span>)</span>
 <b>by </b><i><a class="ref" href="supinf_2.html#T5" title="SUPINF_2:th.5">SUPINF_2:5</a></i>;<br/><a NAME="E112:105_1_2_7_1_3_3"/><b>then </b>
<span class="p1">(<span class="default"><a href="mesfunc5.html#K5" title="MESFUNC5:func.5">integral'</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">g</font></span>)</span> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">e</font> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <span class="p4">(<span class="default"><a href="pre_circ.html#K1" title="PRE_CIRC:func.1">max</a> <font color="Maroon" title="c10">RINGA</font></span>)</span></span>)</span> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <span class="p4">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">g</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="mesfunc5.html#K5" title="MESFUNC5:func.5">integral'</a> <font color="Maroon" title="c3">M</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c25">n</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c27">B</font></span>)</span>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E107:105_1_2_7_1_3_3"><i><font color="Green" title="E209">A228</font></i></a>, <a class="txt" href="mesfunc5.html#E110:105_1_2_7_1_3_3"><i><font color="Green" title="E210">A229</font></i></a>, <a class="txt" href="mesfunc5.html#E54:105_1_2_7_1_3_3"><i><font color="Green" title="E185">A201</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><b>hence </b><a NAME="E113:105_1_2_7_1_3_3"/>
<span class="p1">(<span class="default"><a href="mesfunc5.html#K5" title="MESFUNC5:func.5">integral'</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">g</font></span>)</span> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">e</font> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <span class="p4">(<span class="default"><a href="pre_circ.html#K1" title="PRE_CIRC:func.1">max</a> <font color="Maroon" title="c10">RINGA</font></span>)</span></span>)</span> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <span class="p4">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">g</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="mesfunc5.html#K5" title="MESFUNC5:func.5">integral'</a> <font color="Maroon" title="c3">M</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c25">n</font></span>)</span>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E25:105_1_2_7_1_3_3"><i><font color="Green" title="E164">A173</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E28:105_1_2_7_1_3"/>
for <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Maroon" title="c23">N0</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">n</font> holds <br/><span class="p1">(<span class="default"><a href="mesfunc5.html#K5" title="MESFUNC5:func.5">integral'</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">g</font></span>)</span> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">e</font> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <span class="p4">(<span class="default"><a href="pre_circ.html#K1" title="PRE_CIRC:func.1">max</a> <font color="Maroon" title="c10">RINGA</font></span>)</span></span>)</span> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <span class="p4">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">g</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="mesfunc5.html#K5" title="MESFUNC5:func.5">integral'</a> <font color="Maroon" title="c3">M</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Olive" title="b1">n</font></span>)</span>
 ;<br/>


</div><b>end;</b></div><a NAME="E16:105_1_2_7_1"/><i><font color="Green" title="E134">A230</font></i>: 
for <font color="Olive" title="b1">e</font> being  <a href="supinf_1.html#NM3" title="SUPINF_1:NM.3">R_eal</a>  st 0 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">e</font> &amp; <font color="Olive" title="b1">e</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <span class="p1">(<span class="default"><a href="seq_4.html#K3" title="SEQ_4:func.3">min</a> <font color="Maroon" title="c10">RINGA</font></span>)</span> holds <br/> ex <font color="Olive" title="b2">N0</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <br/>for <font color="Olive" title="b3">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b2">N0</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b3">n</font> holds <br/><span class="p1">(<span class="default"><a href="mesfunc5.html#K5" title="MESFUNC5:func.5">integral'</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">g</font></span>)</span> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">e</font> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <span class="p4">(<span class="default"><a href="pre_circ.html#K1" title="PRE_CIRC:func.1">max</a> <font color="Maroon" title="c10">RINGA</font></span>)</span></span>)</span> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <span class="p4">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">g</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">L</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Olive" title="b3">n</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="105_1_2_7_1_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c18">e</font> be   <a href="supinf_1.html#NM3" title="SUPINF_1:NM.3">R_eal</a>;<br/>

<b>assume </b><a NAME="E1:105_1_2_7_1_4"/>
( 0 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c18">e</font> &amp; <font color="Maroon" title="c18">e</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <span class="p1">(<span class="default"><a href="seq_4.html#K3" title="SEQ_4:func.3">min</a> <font color="Maroon" title="c10">RINGA</font></span>)</span> )
 ;<br/>

<b>then consider </b><font color="Maroon" title="c19">N0</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E3:105_1_2_7_1_4"/><i><font color="Green" title="E135">A231</font></i>: 
for <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Maroon" title="c19">N0</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">n</font> holds <br/><span class="p1">(<span class="default"><a href="mesfunc5.html#K5" title="MESFUNC5:func.5">integral'</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">g</font></span>)</span> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">e</font> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <span class="p4">(<span class="default"><a href="pre_circ.html#K1" title="PRE_CIRC:func.1">max</a> <font color="Maroon" title="c10">RINGA</font></span>)</span></span>)</span> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <span class="p4">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">g</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="mesfunc5.html#K5" title="MESFUNC5:func.5">integral'</a> <font color="Maroon" title="c3">M</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Olive" title="b1">n</font></span>)</span>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E15:105_1_2_7_1"><i><font color="Green" title="E133">A141</font></i></a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="105_1_2_7_1_4_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c20">n</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/><b>assume </b><a NAME="E1:105_1_2_7_1_4_1"/>
<font color="Maroon" title="c19">N0</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c20">n</font>
 ;<br/><a NAME="E2:105_1_2_7_1_4_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="mesfunc5.html#K5" title="MESFUNC5:func.5">integral'</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">g</font></span>)</span> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">e</font> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <span class="p4">(<span class="default"><a href="pre_circ.html#K1" title="PRE_CIRC:func.1">max</a> <font color="Maroon" title="c10">RINGA</font></span>)</span></span>)</span> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <span class="p4">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">g</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="mesfunc5.html#K5" title="MESFUNC5:func.5">integral'</a> <font color="Maroon" title="c3">M</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c20">n</font></span>)</span>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E3:105_1_2_7_1_4"><i><font color="Green" title="E135">A231</font></i></a></i>;<br/><b>hence </b><a NAME="E3:105_1_2_7_1_4_1"/>
<span class="p1">(<span class="default"><a href="mesfunc5.html#K5" title="MESFUNC5:func.5">integral'</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">g</font></span>)</span> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">e</font> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <span class="p4">(<span class="default"><a href="pre_circ.html#K1" title="PRE_CIRC:func.1">max</a> <font color="Maroon" title="c10">RINGA</font></span>)</span></span>)</span> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <span class="p4">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">g</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">L</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c20">n</font>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E8:105"><i><font color="Green" title="E106">A8</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:105_1_2_7_1_4"/>
 ex <font color="Olive" title="b1">N0</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <br/>for <font color="Olive" title="b2">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">N0</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">n</font> holds <br/><span class="p1">(<span class="default"><a href="mesfunc5.html#K5" title="MESFUNC5:func.5">integral'</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">g</font></span>)</span> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">e</font> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <span class="p4">(<span class="default"><a href="pre_circ.html#K1" title="PRE_CIRC:func.1">max</a> <font color="Maroon" title="c10">RINGA</font></span>)</span></span>)</span> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <span class="p4">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">g</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">L</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Olive" title="b2">n</font>
 ;<br/>


</div><b>end;</b></div><a NAME="E17:105_1_2_7_1"/><i><font color="Green" title="E135">A232</font></i>: 
for <font color="Olive" title="b1">e1</font> being  <a href="supinf_1.html#NM3" title="SUPINF_1:NM.3">R_eal</a>  st 0 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">e1</font> holds <br/> ex <font color="Olive" title="b2">e</font> being  <a href="supinf_1.html#NM3" title="SUPINF_1:NM.3">R_eal</a> st <br/>( 0 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b2">e</font> &amp; <font color="Olive" title="b2">e</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <span class="p1">(<span class="default"><a href="seq_4.html#K3" title="SEQ_4:func.3">min</a> <font color="Maroon" title="c10">RINGA</font></span>)</span> &amp; <font color="Olive" title="b2">e</font> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <span class="p3">(<span class="default"><a href="pre_circ.html#K1" title="PRE_CIRC:func.1">max</a> <font color="Maroon" title="c10">RINGA</font></span>)</span></span>)</span> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <span class="p3">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">g</font></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">e1</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="105_1_2_7_1_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c18">e1</font> be   <a href="supinf_1.html#NM3" title="SUPINF_1:NM.3">R_eal</a>;<br/>

<b>assume </b><a NAME="E1:105_1_2_7_1_5"/><i><font color="Green" title="E136">A233</font></i>: 
0 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c18">e1</font>
 ;<br/>

<b>reconsider </b><font color="Maroon" title="c19">ralpha</font> =  <a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <span class="p1">(<span class="default"><a href="seq_4.html#K3" title="SEQ_4:func.3">min</a> <font color="Maroon" title="c10">RINGA</font></span>)</span> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> <b>by </b><i><a class="ref" href="xxreal_0.html#T14" title="XXREAL_0:th.14">XXREAL_0:14</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c20">rdomg</font> = <font color="Maroon" title="c3">M</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">g</font></span>)</span> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> <b>by </b><i><a class="txt" href="mesfunc5.html#E7:105_1_2_7_1"><i><font color="Green" title="E128">A130</font></i></a>, <a class="ref" href="xxreal_0.html#T14" title="XXREAL_0:th.14">XXREAL_0:14</a></i>;<br/>
<a NAME="E4:105_1_2_7_1_5"/>
 <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c16">DG</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T2" title="XBOOLE_1:th.2">XBOOLE_1:2</a></i>;<br/>
<a NAME="E5:105_1_2_7_1_5"/><b>then </b><i><font color="Green" title="E137">A234</font></i>: 
0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c20">rdomg</font>
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E3:105_1_2_7_1"><i><font color="Green" title="E126">A128</font></i></a>, <a class="txt" href="mesfunc5.html#E5:105_1_2_7_1"><i><font color="Green" title="E127">A129</font></i></a>, <a class="ref" href="measure1.html#T62" title="MEASURE1:th.62">MEASURE1:62</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="105_1_2_7_1_5_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:105_1_2_7_1_5_1"/>
( <font color="Maroon" title="c18">e1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>  or <font color="Maroon" title="c18">e1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="105_1_2_7_1_5_1_1"><b>suppose </b></a><a NAME="E1:105_1_2_7_1_5_1_1"/><i><font color="Green" title="E138">A235</font></i>: 
<font color="Maroon" title="c18">e1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a> 
 ;<br/><div class="add"><b>set </b><font color="Maroon" title="c21">z</font> = <font color="Maroon" title="c19">ralpha</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2;<br/><a NAME="E2:105_1_2_7_1_5_1_1"/><i><font color="Green" title="E139">A236</font></i>: 
( 0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c19">ralpha</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2 &amp; <font color="Maroon" title="c19">ralpha</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <span class="p1">(<span class="default"><a href="seq_4.html#K3" title="SEQ_4:func.3">min</a> <font color="Maroon" title="c10">RINGA</font></span>)</span> )
 <b>by </b><i><a class="txt" href="mesfunc5.html#E33:105_1_2"><i><font color="Green" title="E123">A116</font></i></a>, <a class="ref" href="xreal_1.html#T217" title="XREAL_1:th.217">XREAL_1:217</a>, <a class="ref" href="xreal_1.html#T218" title="XREAL_1:th.218">XREAL_1:218</a></i>;<br/><a NAME="E3:105_1_2_7_1_5_1_1"/>
<span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <span class="p2">(<span class="default"><font color="Maroon" title="c19">ralpha</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span></span>)</span> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <span class="p3">(<span class="default"><a href="pre_circ.html#K1" title="PRE_CIRC:func.1">max</a> <font color="Maroon" title="c10">RINGA</font></span>)</span></span>)</span> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <span class="p3">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">g</font></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a> 
 <b>by </b><i><a class="ref" href="xxreal_0.html#T4" title="XXREAL_0:th.4">XXREAL_0:4</a></i>;<br/><b>hence </b><a NAME="E4:105_1_2_7_1_5_1_1"/>
 ex <font color="Olive" title="b1">e</font> being  <a href="supinf_1.html#NM3" title="SUPINF_1:NM.3">R_eal</a> st <br/>( 0 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">e</font> &amp; <font color="Olive" title="b1">e</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <span class="p1">(<span class="default"><a href="seq_4.html#K3" title="SEQ_4:func.3">min</a> <font color="Maroon" title="c10">RINGA</font></span>)</span> &amp; <font color="Olive" title="b1">e</font> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <span class="p3">(<span class="default"><a href="pre_circ.html#K1" title="PRE_CIRC:func.1">max</a> <font color="Maroon" title="c10">RINGA</font></span>)</span></span>)</span> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <span class="p3">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">g</font></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c18">e1</font> )
 <b>by </b><i><a class="txt" href="mesfunc5.html#E2:105_1_2_7_1_5_1_1"><i><font color="Green" title="E139">A236</font></i></a>, <a class="txt" href="mesfunc5.html#E1:105_1_2_7_1_5_1_1"><i><font color="Green" title="E138">A235</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="105_1_2_7_1_5_1_2"><b>suppose </b></a><a NAME="E1:105_1_2_7_1_5_1_2"/>
<font color="Maroon" title="c18">e1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a> 
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon" title="c21">re1</font> = <font color="Maroon" title="c18">e1</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> <b>by </b><i><a class="txt" href="mesfunc5.html#E1:105_1_2_7_1_5"><i><font color="Green" title="E136">A233</font></i></a>, <a class="ref" href="xxreal_0.html#T14" title="XXREAL_0:th.14">XXREAL_0:14</a></i>;<br/><a NAME="E3:105_1_2_7_1_5_1_2"/><i><font color="Green" title="E138">A237</font></i>: 
0 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c13">beta1</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <font color="Maroon" title="c20">rdomg</font>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E5:105_1_2_7_1_5"><i><font color="Green" title="E137">A234</font></i></a>, <a class="txt" href="mesfunc5.html#E10:105_1_2_7_1"><i><font color="Green" title="E130">A132</font></i></a>, <a class="ref" href="xreal_1.html#T10" title="XREAL_1:th.10">XREAL_1:10</a></i>;<br/><b>set </b><font color="Maroon" title="c22">x</font> = <font color="Maroon" title="c21">re1</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">beta1</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <font color="Maroon" title="c20">rdomg</font></span>)</span>;<br/><b>set </b><font color="Maroon" title="c23">y</font> = <font color="Maroon" title="c19">ralpha</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2;<br/><a NAME="E4:105_1_2_7_1_5_1_2"/><i><font color="Green" title="E139">A238</font></i>: 
<font color="Maroon" title="c19">ralpha</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c19">ralpha</font>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E33:105_1_2"><i><font color="Green" title="E123">A116</font></i></a>, <a class="ref" href="xreal_1.html#T218" title="XREAL_1:th.218">XREAL_1:218</a></i>;<br/><b>set </b><font color="Maroon" title="c24">z</font> =  <a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">re1</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">beta1</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <font color="Maroon" title="c20">rdomg</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c19">ralpha</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span>;<br/><div><i><font color="Green" title="E140">A239</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="105_1_2_7_1_5_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="105_1_2_7_1_5_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:105_1_2_7_1_5_1_2_1_1"/>
(  <a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">re1</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">beta1</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <font color="Maroon" title="c20">rdomg</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c19">ralpha</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">re1</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">beta1</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <font color="Maroon" title="c20">rdomg</font></span>)</span> or  <a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">re1</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">beta1</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <font color="Maroon" title="c20">rdomg</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c19">ralpha</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">ralpha</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2 )
 </span><b>by </b><i><a class="ref" href="xxreal_0.html#T15" title="XXREAL_0:th.15">XXREAL_0:15</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="105_1_2_7_1_5_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:105_1_2_7_1_5_1_2_1_1_1"/>
 <a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">re1</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">beta1</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <font color="Maroon" title="c20">rdomg</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c19">ralpha</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">re1</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">beta1</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <font color="Maroon" title="c20">rdomg</font></span>)</span>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:105_1_2_7_1_5_1_2_1_1_1"/>
0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a>  <a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">re1</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">beta1</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <font color="Maroon" title="c20">rdomg</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c19">ralpha</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E3:105_1_2_7_1_5_1_2"><i><font color="Green" title="E138">A237</font></i></a>, <a class="txt" href="mesfunc5.html#E1:105_1_2_7_1_5"><i><font color="Green" title="E136">A233</font></i></a>, <a class="ref" href="xreal_1.html#T141" title="XREAL_1:th.141">XREAL_1:141</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="105_1_2_7_1_5_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:105_1_2_7_1_5_1_2_1_1_2"/>
 <a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">re1</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">beta1</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <font color="Maroon" title="c20">rdomg</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c19">ralpha</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">ralpha</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2
 ;<br/><div class="add"><b>hence </b><a NAME="E2:105_1_2_7_1_5_1_2_1_1_2"/>
0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a>  <a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">re1</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">beta1</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <font color="Maroon" title="c20">rdomg</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c19">ralpha</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E33:105_1_2"><i><font color="Green" title="E123">A116</font></i></a>, <a class="ref" href="xreal_1.html#T217" title="XREAL_1:th.217">XREAL_1:217</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><a NAME="E6:105_1_2_7_1_5_1_2"/>
 <a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">re1</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">beta1</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <font color="Maroon" title="c20">rdomg</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c19">ralpha</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c19">ralpha</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2
 <b>by </b><i><a class="ref" href="xxreal_0.html#T17" title="XXREAL_0:th.17">XXREAL_0:17</a></i>;<br/><a NAME="E7:105_1_2_7_1_5_1_2"/><b>then </b><i><font color="Green" title="E141">A240</font></i>: 
 <a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">re1</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">beta1</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <font color="Maroon" title="c20">rdomg</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c19">ralpha</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c19">ralpha</font>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E4:105_1_2_7_1_5_1_2"><i><font color="Green" title="E139">A238</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E8:105_1_2_7_1_5_1_2"/>
 <a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">re1</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">beta1</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <font color="Maroon" title="c20">rdomg</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c19">ralpha</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c21">re1</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">beta1</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <font color="Maroon" title="c20">rdomg</font></span>)</span>
 <b>by </b><i><a class="ref" href="xxreal_0.html#T17" title="XXREAL_0:th.17">XXREAL_0:17</a></i>;<br/><a NAME="E9:105_1_2_7_1_5_1_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p2">(<span class="default"><font color="Maroon" title="c21">re1</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p3">(<span class="default"><font color="Maroon" title="c13">beta1</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <font color="Maroon" title="c20">rdomg</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c19">ralpha</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">beta1</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <font color="Maroon" title="c20">rdomg</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">re1</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">beta1</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <font color="Maroon" title="c20">rdomg</font></span>)</span></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">beta1</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <font color="Maroon" title="c20">rdomg</font></span>)</span>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E3:105_1_2_7_1_5_1_2"><i><font color="Green" title="E138">A237</font></i></a>, <a class="ref" href="xreal_1.html#T66" title="XREAL_1:th.66">XREAL_1:66</a></i>;<br/><a NAME="E10:105_1_2_7_1_5_1_2"/><b>then </b><i><font color="Green" title="E142">A241</font></i>: 
<span class="p1">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p2">(<span class="default"><font color="Maroon" title="c21">re1</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p3">(<span class="default"><font color="Maroon" title="c13">beta1</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <font color="Maroon" title="c20">rdomg</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c19">ralpha</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">beta1</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <font color="Maroon" title="c20">rdomg</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c21">re1</font>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E3:105_1_2_7_1_5_1_2"><i><font color="Green" title="E138">A237</font></i></a>, <a class="ref" href="xcmplx_1.html#T88" title="XCMPLX_1:th.88">XCMPLX_1:88</a></i>;<br/><a NAME="E11:105_1_2_7_1_5_1_2"/>
 <a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">beta1</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <font color="Maroon" title="c20">rdomg</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c13">beta1</font></span>)</span> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c20">rdomg</font></span>)</span>
 <b>by </b><i><a class="ref" href="supinf_2.html#T1" title="SUPINF_2:th.1">SUPINF_2:1</a></i>;<br/><a NAME="E12:105_1_2_7_1_5_1_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p2">(<span class="default"><font color="Maroon" title="c21">re1</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p3">(<span class="default"><font color="Maroon" title="c13">beta1</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <font color="Maroon" title="c20">rdomg</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c19">ralpha</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">beta1</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <font color="Maroon" title="c20">rdomg</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <span class="p2">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p3">(<span class="default"><font color="Maroon" title="c21">re1</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p4">(<span class="default"><font color="Maroon" title="c13">beta1</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <font color="Maroon" title="c20">rdomg</font></span>)</span></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c19">ralpha</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span></span>)</span></span>)</span> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <span class="p3">(<span class="default"><a href="pre_circ.html#K1" title="PRE_CIRC:func.1">max</a> <font color="Maroon" title="c10">RINGA</font></span>)</span></span>)</span> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <span class="p3">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">g</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="extreal1.html#T38" title="EXTREAL1:th.38">EXTREAL1:38</a></i>;<br/><b>hence </b><a NAME="E13:105_1_2_7_1_5_1_2"/>
 ex <font color="Olive" title="b1">e</font> being  <a href="supinf_1.html#NM3" title="SUPINF_1:NM.3">R_eal</a> st <br/>( 0 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">e</font> &amp; <font color="Olive" title="b1">e</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <span class="p1">(<span class="default"><a href="seq_4.html#K3" title="SEQ_4:func.3">min</a> <font color="Maroon" title="c10">RINGA</font></span>)</span> &amp; <font color="Olive" title="b1">e</font> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <span class="p3">(<span class="default"><a href="pre_circ.html#K1" title="PRE_CIRC:func.1">max</a> <font color="Maroon" title="c10">RINGA</font></span>)</span></span>)</span> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <span class="p3">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">g</font></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c18">e1</font> )
 <b>by </b><i><a class="txt" href="mesfunc5.html#E5:105_1_2_7_1_5_1_2"><i><font color="Green" title="E140">A239</font></i></a>, <a class="txt" href="mesfunc5.html#E7:105_1_2_7_1_5_1_2"><i><font color="Green" title="E141">A240</font></i></a>, <a class="txt" href="mesfunc5.html#E10:105_1_2_7_1_5_1_2"><i><font color="Green" title="E142">A241</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><a NAME="E18:105_1_2_7_1"/><i><font color="Green" title="E136">A242</font></i>: 
for <font color="Olive" title="b1">e1</font> being  <a href="supinf_1.html#NM3" title="SUPINF_1:NM.3">R_eal</a>  st 0 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">e1</font> holds <br/> ex <font color="Olive" title="b2">N0</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <br/>for <font color="Olive" title="b3">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b2">N0</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b3">n</font> holds <br/><span class="p1">(<span class="default"><a href="mesfunc5.html#K5" title="MESFUNC5:func.5">integral'</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">g</font></span>)</span> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <font color="Olive" title="b1">e1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">L</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Olive" title="b3">n</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="105_1_2_7_1_6"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c18">e1</font> be   <a href="supinf_1.html#NM3" title="SUPINF_1:NM.3">R_eal</a>;<br/>

<b>assume </b><a NAME="E1:105_1_2_7_1_6"/>
0 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c18">e1</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c19">e</font> being   <a href="supinf_1.html#NM3" title="SUPINF_1:NM.3">R_eal</a><b> such that </b><br/><a NAME="E3:105_1_2_7_1_6"/><i><font color="Green" title="E137">A243</font></i>: 
( 0 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c19">e</font> &amp; <font color="Maroon" title="c19">e</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <span class="p1">(<span class="default"><a href="seq_4.html#K3" title="SEQ_4:func.3">min</a> <font color="Maroon" title="c10">RINGA</font></span>)</span> &amp; <font color="Maroon" title="c19">e</font> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <span class="p3">(<span class="default"><a href="pre_circ.html#K1" title="PRE_CIRC:func.1">max</a> <font color="Maroon" title="c10">RINGA</font></span>)</span></span>)</span> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <span class="p3">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">g</font></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c18">e1</font> )
 <b>by </b><i><a class="txt" href="mesfunc5.html#E17:105_1_2_7_1"><i><font color="Green" title="E135">A232</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c20">N0</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E5:105_1_2_7_1_6"/><i><font color="Green" title="E138">A244</font></i>: 
for <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Maroon" title="c20">N0</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">n</font> holds <br/><span class="p1">(<span class="default"><a href="mesfunc5.html#K5" title="MESFUNC5:func.5">integral'</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">g</font></span>)</span> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">e</font> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <span class="p4">(<span class="default"><a href="pre_circ.html#K1" title="PRE_CIRC:func.1">max</a> <font color="Maroon" title="c10">RINGA</font></span>)</span></span>)</span> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <span class="p4">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">g</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">L</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Olive" title="b1">n</font>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E16:105_1_2_7_1"><i><font color="Green" title="E134">A230</font></i></a>, <a class="txt" href="mesfunc5.html#E3:105_1_2_7_1_6"><i><font color="Green" title="E137">A243</font></i></a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c20">N0</font>
;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="105_1_2_7_1_6_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c21">n</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/><b>assume </b><a NAME="E1:105_1_2_7_1_6_1"/>
<font color="Maroon" title="c20">N0</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c21">n</font>
 ;<br/><a NAME="E2:105_1_2_7_1_6_1"/><b>then </b><i><font color="Green" title="E139">A245</font></i>: 
<span class="p1">(<span class="default"><a href="mesfunc5.html#K5" title="MESFUNC5:func.5">integral'</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">g</font></span>)</span> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">e</font> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <span class="p4">(<span class="default"><a href="pre_circ.html#K1" title="PRE_CIRC:func.1">max</a> <font color="Maroon" title="c10">RINGA</font></span>)</span></span>)</span> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <span class="p4">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">g</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">L</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c21">n</font>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E5:105_1_2_7_1_6"><i><font color="Green" title="E138">A244</font></i></a></i>;<br/><a NAME="E3:105_1_2_7_1_6_1"/>
<span class="p1">(<span class="default"><a href="mesfunc5.html#K5" title="MESFUNC5:func.5">integral'</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">g</font></span>)</span> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <font color="Maroon" title="c18">e1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="mesfunc5.html#K5" title="MESFUNC5:func.5">integral'</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">g</font></span>)</span> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">e</font> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <span class="p4">(<span class="default"><a href="pre_circ.html#K1" title="PRE_CIRC:func.1">max</a> <font color="Maroon" title="c10">RINGA</font></span>)</span></span>)</span> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <span class="p4">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">g</font></span>)</span></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="supinf_2.html#T15" title="SUPINF_2:th.15">SUPINF_2:15</a>, <a class="txt" href="mesfunc5.html#E14:105_1_2_7_1"><i><font color="Green" title="E132">A140</font></i></a>, <a class="txt" href="mesfunc5.html#E3:105_1_2_7_1_6"><i><font color="Green" title="E137">A243</font></i></a></i>;<br/><b>hence </b><a NAME="E4:105_1_2_7_1_6_1"/>
<span class="p1">(<span class="default"><a href="mesfunc5.html#K5" title="MESFUNC5:func.5">integral'</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">g</font></span>)</span> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <font color="Maroon" title="c18">e1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">L</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c21">n</font>
 <b>by </b><i><a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a>, <a class="txt" href="mesfunc5.html#E2:105_1_2_7_1_6_1"><i><font color="Green" title="E139">A245</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E7:105_1_2_7_1_6"/>
for <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Maroon" title="c20">N0</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">n</font> holds <br/><span class="p1">(<span class="default"><a href="mesfunc5.html#K5" title="MESFUNC5:func.5">integral'</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">g</font></span>)</span> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <font color="Maroon" title="c18">e1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">L</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Olive" title="b1">n</font>
 ;<br/>


</div><b>end;</b></div><a NAME="E19:105_1_2_7_1"/><i><font color="Green" title="E137">A246</font></i>: 
for <font color="Olive" title="b1">n</font>, <font color="Olive" title="b2">m</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">m</font> holds <br/><font color="Maroon" title="c6">L</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">L</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Olive" title="b2">m</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="105_1_2_7_1_7"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c18">n</font>, <font color="Maroon" title="c19">m</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:105_1_2_7_1_7"/><i><font color="Green" title="E138">A247</font></i>: 
<font color="Maroon" title="c18">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c19">m</font>
 ;<br/>

<a NAME="E2:105_1_2_7_1_7"/><i><font color="Green" title="E139">A248</font></i>: 
(  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c18">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">g</font> &amp;  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c19">m</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">g</font> )
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E4:105"><i><font color="Green" title="E102">A4</font></i></a></i>;<br/>
<a NAME="E3:105_1_2_7_1_7"/><i><font color="Green" title="E140">A249</font></i>: 
( <font color="Maroon" title="c5">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c18">n</font> is <a href="supinf_2.html#V6" title="SUPINF_2:attr.6">nonnegative</a> &amp; <font color="Maroon" title="c5">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c19">m</font> is <a href="supinf_2.html#V6" title="SUPINF_2:attr.6">nonnegative</a> )
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E5:105"><i><font color="Green" title="E103">A5</font></i></a></i>;<br/>
<a NAME="E4:105_1_2_7_1_7"/><i><font color="Green" title="E141">A250</font></i>: 
( <font color="Maroon" title="c5">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c18">n</font> <a href="mesfunc2.html#R1" title="MESFUNC2:pred.1">is_simple_func_in</a> <font color="Maroon" title="c2">S</font> &amp; <font color="Maroon" title="c5">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c19">m</font> <a href="mesfunc2.html#R1" title="MESFUNC2:pred.1">is_simple_func_in</a> <font color="Maroon" title="c2">S</font> )
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E3:105"><i><font color="Green" title="E101">A3</font></i></a></i>;<br/>
<a NAME="E5:105_1_2_7_1_7"/><i><font color="Green" title="E142">A251</font></i>: 
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c19">m</font></span>)</span> <a href="mesfunc1.html#K6" title="MESFUNC1:func.6">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c18">n</font></span>)</span></span>)</span> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c18">n</font></span>)</span> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Olive" title="b1">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c19">m</font></span>)</span> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Olive" title="b1">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="105_1_2_7_1_7_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c20">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:105_1_2_7_1_7_1"/>
<font color="Maroon" title="c20">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c19">m</font></span>)</span> <a href="mesfunc1.html#K6" title="MESFUNC1:func.6">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c18">n</font></span>)</span></span>)</span>
 ;<br/>

<a NAME="E2:105_1_2_7_1_7_1"/><b>then </b>
<font color="Maroon" title="c20">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c19">m</font></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c18">n</font></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c19">m</font></span>)</span> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p4"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c18">n</font></span>)</span> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p4"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c19">m</font></span>)</span> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p4"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c18">n</font></span>)</span> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p4"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="mesfunc1.html#D4" title="MESFUNC1:def.4">MESFUNC1:def 4</a></i>;<br/>
<a NAME="E3:105_1_2_7_1_7_1"/><b>then </b>
<font color="Maroon" title="c20">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c19">m</font></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c18">n</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/>
<b>hence </b><a NAME="E4:105_1_2_7_1_7_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c18">n</font></span>)</span> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c20">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c19">m</font></span>)</span> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c20">x</font>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E1:105_1_2_7_1_7"><i><font color="Green" title="E138">A247</font></i></a>, <a class="txt" href="mesfunc5.html#E6:105"><i><font color="Green" title="E104">A6</font></i></a>, <a class="txt" href="mesfunc5.html#E2:105_1_2_7_1_7"><i><font color="Green" title="E139">A248</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E6:105_1_2_7_1_7"/><b>then </b><i><font color="Green" title="E143">A252</font></i>: 
 <a href="mesfunc5.html#K5" title="MESFUNC5:func.5">integral'</a> <font color="Maroon" title="c3">M</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c18">n</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c19">m</font></span>)</span> <a href="mesfunc1.html#K6" title="MESFUNC1:func.6">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c18">n</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="mesfunc5.html#K5" title="MESFUNC5:func.5">integral'</a> <font color="Maroon" title="c3">M</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c19">m</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c19">m</font></span>)</span> <a href="mesfunc1.html#K6" title="MESFUNC1:func.6">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c18">n</font></span>)</span></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="mesfunc5.html#T76" target="_self" title="MESFUNC5:th.76">Th76</a>, <a class="txt" href="mesfunc5.html#E4:105_1_2_7_1_7"><i><font color="Green" title="E141">A250</font></i></a>, <a class="txt" href="mesfunc5.html#E3:105_1_2_7_1_7"><i><font color="Green" title="E140">A249</font></i></a></i>;<br/>
<a NAME="E7:105_1_2_7_1_7"/>
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c19">m</font></span>)</span> <a href="mesfunc1.html#K6" title="MESFUNC1:func.6">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c18">n</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c19">m</font></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c18">n</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E4:105_1_2_7_1_7"><i><font color="Green" title="E141">A250</font></i></a>, <a class="txt" href="mesfunc5.html#E3:105_1_2_7_1_7"><i><font color="Green" title="E140">A249</font></i></a>, <a class="txt" href="mesfunc5.html#E5:105_1_2_7_1_7"><i><font color="Green" title="E142">A251</font></i></a>, <a class="ref" href="mesfunc5.html#T75" target="_self" title="MESFUNC5:th.75">Th75</a></i>;<br/>
<a NAME="E8:105_1_2_7_1_7"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c19">m</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c19">m</font></span>)</span> <a href="mesfunc1.html#K6" title="MESFUNC1:func.6">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c18">n</font></span>)</span></span>)</span></span>)</span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c5">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c19">m</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c18">n</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c19">m</font></span>)</span> <a href="mesfunc1.html#K6" title="MESFUNC1:func.6">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c18">n</font></span>)</span></span>)</span></span>)</span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c5">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c18">n</font> )
 
<b>by </b><i><a class="ref" href="grfunc_1.html#T64" title="GRFUNC_1:th.64">GRFUNC_1:64</a>, <a class="txt" href="mesfunc5.html#E2:105_1_2_7_1_7"><i><font color="Green" title="E139">A248</font></i></a></i>;<br/>
<a NAME="E9:105_1_2_7_1_7"/><b>then </b>
<font color="Maroon" title="c6">L</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c18">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="mesfunc5.html#K5" title="MESFUNC5:func.5">integral'</a> <font color="Maroon" title="c3">M</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c19">m</font></span>)</span>
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E8:105"><i><font color="Green" title="E106">A8</font></i></a>, <a class="txt" href="mesfunc5.html#E6:105_1_2_7_1_7"><i><font color="Green" title="E143">A252</font></i></a></i>;<br/>
<b>hence </b><a NAME="E10:105_1_2_7_1_7"/>
<font color="Maroon" title="c6">L</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c18">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">L</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c19">m</font>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E8:105"><i><font color="Green" title="E106">A8</font></i></a></i>;<br/>


</div><b>end;</b></div><a NAME="E20:105_1_2_7_1"/><i><font color="Green" title="E138">A253</font></i>: 
for <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">L</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Olive" title="b1">n</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="105_1_2_7_1_8"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c18">n</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/>

<a NAME="E1:105_1_2_7_1_8"/><i><font color="Green" title="E139">A254</font></i>: 
<font color="Maroon" title="c5">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c18">n</font> is <a href="supinf_2.html#V6" title="SUPINF_2:attr.6">nonnegative</a>
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E5:105"><i><font color="Green" title="E103">A5</font></i></a></i>;<br/>
<a NAME="E2:105_1_2_7_1_8"/>
<font color="Maroon" title="c5">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c18">n</font> <a href="mesfunc2.html#R1" title="MESFUNC2:pred.1">is_simple_func_in</a> <font color="Maroon" title="c2">S</font>
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E3:105"><i><font color="Green" title="E101">A3</font></i></a></i>;<br/>
<a NAME="E3:105_1_2_7_1_8"/><b>then </b>
0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="mesfunc5.html#K5" title="MESFUNC5:func.5">integral'</a> <font color="Maroon" title="c3">M</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c18">n</font></span>)</span>
 
<b>by </b><i><a class="ref" href="mesfunc5.html#T74" target="_self" title="MESFUNC5:th.74">Th74</a>, <a class="txt" href="mesfunc5.html#E1:105_1_2_7_1_8"><i><font color="Green" title="E139">A254</font></i></a></i>;<br/>
<b>hence </b><a NAME="E4:105_1_2_7_1_8"/>
0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">L</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c18">n</font>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E8:105"><i><font color="Green" title="E106">A8</font></i></a></i>;<br/>


</div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="105_1_2_7_1_9"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:105_1_2_7_1_9"/>
(  ex <font color="Olive" title="b1">K</font> being <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>  st <br/>( 0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Olive" title="b1">K</font> &amp; ( for <font color="Olive" title="b2">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  holds <font color="Maroon" title="c6">L</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Olive" title="b2">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">K</font> ) ) or for <font color="Olive" title="b1">K</font> being <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>   holds <br/>(  not 0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Olive" title="b1">K</font> or  ex <font color="Olive" title="b2">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st not <font color="Maroon" title="c6">L</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Olive" title="b2">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">K</font> ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="105_1_2_7_1_9_1"><b>suppose </b></a><a NAME="E1:105_1_2_7_1_9_1"/>
 ex <font color="Olive" title="b1">K</font> being <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>  st <br/>( 0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Olive" title="b1">K</font> &amp; ( for <font color="Olive" title="b2">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  holds <font color="Maroon" title="c6">L</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Olive" title="b2">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">K</font> ) )
 ;<br/><div class="add"><b>then consider </b><font color="Maroon" title="c18">K</font> being  <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> <b> such that </b><br/><a NAME="E3:105_1_2_7_1_9_1"/>
0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c18">K</font>
 <b>and </b><br/><a NAME="E4:105_1_2_7_1_9_1"/><i><font color="Green" title="E139">A255</font></i>: 
for <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  holds <font color="Maroon" title="c6">L</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c18">K</font>
 ;<br/><a NAME="E5:105_1_2_7_1_9_1"/><i><font color="Green" title="E140">A256</font></i>: 
for <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <font color="Maroon" title="c6">L</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="supinf_2.html#K8" title="SUPINF_2:func.8">sup</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K19" title="SUPINF_2:func.19">rng</a> <font color="Maroon" title="c6">L</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="105_1_2_7_1_9_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c19">n</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/>

<a NAME="E1:105_1_2_7_1_9_1_1"/><i><font color="Green" title="E141">A257</font></i>: 
 <a href="supinf_2.html#K8" title="SUPINF_2:func.8">sup</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K19" title="SUPINF_2:func.19">rng</a> <font color="Maroon" title="c6">L</font></span>)</span> is    <a href="supinf_1.html#M1" title="SUPINF_1:mode.1">majorant</a> of  <a href="supinf_2.html#K19" title="SUPINF_2:func.19">rng</a> <font color="Maroon" title="c6">L</font>
 
<b>by </b><i><a class="ref" href="supinf_1.html#D16" title="SUPINF_1:def.16">SUPINF_1:def 16</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c20">n</font> = <font color="Maroon" title="c19">n</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/>
<a NAME="E3:105_1_2_7_1_9_1_1"/>
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c6">L</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E4:105_1_2_7_1_9_1_1"/><b>then </b>
<font color="Maroon" title="c6">L</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c20">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="supinf_2.html#K19" title="SUPINF_2:func.19">rng</a> <font color="Maroon" title="c6">L</font>
 
<b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<b>hence </b><a NAME="E5:105_1_2_7_1_9_1_1"/>
<font color="Maroon" title="c6">L</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c19">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="supinf_2.html#K8" title="SUPINF_2:func.8">sup</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K19" title="SUPINF_2:func.19">rng</a> <font color="Maroon" title="c6">L</font></span>)</span>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E1:105_1_2_7_1_9_1_1"><i><font color="Green" title="E141">A257</font></i></a>, <a class="ref" href="supinf_1.html#D9" title="SUPINF_1:def.9">SUPINF_1:def 9</a></i>;<br/>


</div><b>end;</b></div><a NAME="E6:105_1_2_7_1_9_1"/><b>then </b>
<font color="Maroon" title="c6">L</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="supinf_2.html#K8" title="SUPINF_2:func.8">sup</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K19" title="SUPINF_2:func.19">rng</a> <font color="Maroon" title="c6">L</font></span>)</span>
 ;<br/><a NAME="E7:105_1_2_7_1_9_1"/><b>then </b><i><font color="Green" title="E141">A258</font></i>: 
 <a href="supinf_2.html#K8" title="SUPINF_2:func.8">sup</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K19" title="SUPINF_2:func.19">rng</a> <font color="Maroon" title="c6">L</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a> 
 <b>by </b><i><a class="txt" href="mesfunc5.html#E20:105_1_2_7_1"><i><font color="Green" title="E138">A253</font></i></a></i>;<br/><a NAME="E8:105_1_2_7_1_9_1"/>
 <a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c18">K</font> is    <a href="supinf_1.html#M1" title="SUPINF_1:mode.1">majorant</a> of  <a href="supinf_2.html#K19" title="SUPINF_2:func.19">rng</a> <font color="Maroon" title="c6">L</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="105_1_2_7_1_9_1_2"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="105_1_2_7_1_9_1_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c19">x</font> be  <a href="xxreal_0.html#V1" title="XXREAL_0:attr.1">ext-real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ;<br/><b>assume </b><a NAME="E1:105_1_2_7_1_9_1_2_1"/>
<font color="Maroon" title="c19">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="supinf_2.html#K19" title="SUPINF_2:func.19">rng</a> <font color="Maroon" title="c6">L</font>
 ;<br/><b>then consider </b><font color="Maroon" title="c20">z</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:105_1_2_7_1_9_1_2_1"/><i><font color="Green" title="E142">A259</font></i>: 
<font color="Maroon" title="c20">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c6">L</font>
 <b>and </b><br/><a NAME="E4:105_1_2_7_1_9_1_2_1"/><i><font color="Green" title="E143">A260</font></i>: 
<font color="Maroon" title="c19">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">L</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c20">z</font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><a NAME="E5:105_1_2_7_1_9_1_2_1"/>
<font color="Maroon" title="c20">z</font> is   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E3:105_1_2_7_1_9_1_2_1"><i><font color="Green" title="E142">A259</font></i></a>, <a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/><b>hence </b><a NAME="E6:105_1_2_7_1_9_1_2_1"/>
<font color="Maroon" title="c19">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c18">K</font>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E4:105_1_2_7_1_9_1"><i><font color="Green" title="E139">A255</font></i></a>, <a class="txt" href="mesfunc5.html#E4:105_1_2_7_1_9_1_2_1"><i><font color="Green" title="E143">A260</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:105_1_2_7_1_9_1_2"/>
 <a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c18">K</font> is    <a href="supinf_1.html#M1" title="SUPINF_1:mode.1">majorant</a> of  <a href="supinf_2.html#K19" title="SUPINF_2:func.19">rng</a> <font color="Maroon" title="c6">L</font>
 <b>by </b><i><a class="ref" href="supinf_1.html#D9" title="SUPINF_1:def.9">SUPINF_1:def 9</a></i>;<br/>


</div><b>end;</b></div><a NAME="E9:105_1_2_7_1_9_1"/><b>then </b><i><font color="Green" title="E142">A261</font></i>: 
 <a href="supinf_2.html#K8" title="SUPINF_2:func.8">sup</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K19" title="SUPINF_2:func.19">rng</a> <font color="Maroon" title="c6">L</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c18">K</font>
 <b>by </b><i><a class="ref" href="supinf_1.html#D16" title="SUPINF_1:def.16">SUPINF_1:def 16</a></i>;<br/><a NAME="E10:105_1_2_7_1_9_1"/>
 <a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c18">K</font> is   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>
 <b>by </b><i><a class="ref" href="xreal_0.html#D1" title="XREAL_0:def.1">XREAL_0:def 1</a></i>;<br/><a NAME="E11:105_1_2_7_1_9_1"/><b>then </b><i><font color="Green" title="E143">A262</font></i>: 
 <a href="supinf_2.html#K8" title="SUPINF_2:func.8">sup</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K19" title="SUPINF_2:func.19">rng</a> <font color="Maroon" title="c6">L</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a> 
 <b>by </b><i><a class="txt" href="mesfunc5.html#E9:105_1_2_7_1_9_1"><i><font color="Green" title="E142">A261</font></i></a>, <a class="ref" href="xxreal_0.html#T9" title="XXREAL_0:th.9">XXREAL_0:9</a></i>;<br/><a NAME="E12:105_1_2_7_1_9_1"/><i><font color="Green" title="E144">A263</font></i>: 
for <font color="Olive" title="b1">p</font> being <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>   st 0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Olive" title="b1">p</font> holds <br/> ex <font color="Olive" title="b2">N0</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <br/>for <font color="Olive" title="b3">m</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b2">N0</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b3">m</font> holds <br/><span class="p1"><a href="extreal1.html#K4" title="EXTREAL1:func.4">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">L</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Olive" title="b3">m</font></span>)</span> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <span class="p3">(<span class="default"><a href="supinf_2.html#K8" title="SUPINF_2:func.8">sup</a> <span class="p4">(<span class="default"><a href="supinf_2.html#K19" title="SUPINF_2:func.19">rng</a> <font color="Maroon" title="c6">L</font></span>)</span></span>)</span></span>)</span></span><a href="extreal1.html#K4" title="EXTREAL1:func.4">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">p</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="105_1_2_7_1_9_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c19">p</font> be  <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ;<br/>

<b>assume </b><a NAME="E1:105_1_2_7_1_9_1_3"/><i><font color="Green" title="E145">A264</font></i>: 
0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c19">p</font>
 ;<br/>

<a NAME="E2:105_1_2_7_1_9_1_3"/>
 <a href="supinf_2.html#K8" title="SUPINF_2:func.8">sup</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K19" title="SUPINF_2:func.19">rng</a> <font color="Maroon" title="c6">L</font></span>)</span> is   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E7:105_1_2_7_1_9_1"><i><font color="Green" title="E141">A258</font></i></a>, <a class="txt" href="mesfunc5.html#E11:105_1_2_7_1_9_1"><i><font color="Green" title="E143">A262</font></i></a>, <a class="ref" href="xxreal_0.html#T14" title="XXREAL_0:th.14">XXREAL_0:14</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c20">y</font> being  <a href="xxreal_0.html#V1" title="XXREAL_0:attr.1">ext-real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> <b> such that </b><br/><a NAME="E4:105_1_2_7_1_9_1_3"/><i><font color="Green" title="E146">A265</font></i>: 
( <font color="Maroon" title="c20">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="supinf_2.html#K19" title="SUPINF_2:func.19">rng</a> <font color="Maroon" title="c6">L</font> &amp; <span class="p1">(<span class="default"><a href="supinf_2.html#K8" title="SUPINF_2:func.8">sup</a> <span class="p2">(<span class="default"><a href="supinf_2.html#K19" title="SUPINF_2:func.19">rng</a> <font color="Maroon" title="c6">L</font></span>)</span></span>)</span> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c19">p</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c20">y</font> )
 <b>by </b><i><a class="txt" href="mesfunc5.html#E1:105_1_2_7_1_9_1_3"><i><font color="Green" title="E145">A264</font></i></a>, <a class="ref" href="measure6.html#T33" title="MEASURE6:th.33">MEASURE6:33</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c21">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E6:105_1_2_7_1_9_1_3"/><i><font color="Green" title="E147">A266</font></i>: 
( <font color="Maroon" title="c21">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c6">L</font> &amp; <font color="Maroon" title="c20">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">L</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c21">x</font> )
 <b>by </b><i><a class="txt" href="mesfunc5.html#E4:105_1_2_7_1_9_1_3"><i><font color="Green" title="E146">A265</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c22">N0</font> = <font color="Maroon" title="c21">x</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="txt" href="mesfunc5.html#E6:105_1_2_7_1_9_1_3"><i><font color="Green" title="E147">A266</font></i></a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c22">N0</font>
;<br/>

<b>let </b><font color="Maroon" title="c23">n</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/>

<b>assume </b><a NAME="E8:105_1_2_7_1_9_1_3"/>
<font color="Maroon" title="c22">N0</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c23">n</font>
 ;<br/>

<a NAME="E9:105_1_2_7_1_9_1_3"/><b>then </b>
<font color="Maroon" title="c6">L</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c22">N0</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">L</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c23">n</font>
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E19:105_1_2_7_1"><i><font color="Green" title="E137">A246</font></i></a></i>;<br/>
<a NAME="E10:105_1_2_7_1_9_1_3"/><b>then </b><i><font color="Green" title="E148">A267</font></i>: 
<span class="p1">(<span class="default"><a href="supinf_2.html#K8" title="SUPINF_2:func.8">sup</a> <span class="p2">(<span class="default"><a href="supinf_2.html#K19" title="SUPINF_2:func.19">rng</a> <font color="Maroon" title="c6">L</font></span>)</span></span>)</span> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c19">p</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">L</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c23">n</font>
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E6:105_1_2_7_1_9_1_3"><i><font color="Green" title="E147">A266</font></i></a>, <a class="txt" href="mesfunc5.html#E4:105_1_2_7_1_9_1_3"><i><font color="Green" title="E146">A265</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E11:105_1_2_7_1_9_1_3"/><i><font color="Green" title="E149">A268</font></i>: 
<font color="Maroon" title="c6">L</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c23">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="supinf_2.html#K8" title="SUPINF_2:func.8">sup</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K19" title="SUPINF_2:func.19">rng</a> <font color="Maroon" title="c6">L</font></span>)</span>
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E5:105_1_2_7_1_9_1"><i><font color="Green" title="E140">A256</font></i></a></i>;<br/>
<a NAME="E12:105_1_2_7_1_9_1_3"/>
<span class="p1">(<span class="default"><a href="supinf_2.html#K8" title="SUPINF_2:func.8">sup</a> <span class="p2">(<span class="default"><a href="supinf_2.html#K19" title="SUPINF_2:func.19">rng</a> <font color="Maroon" title="c6">L</font></span>)</span></span>)</span> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K8" title="SUPINF_2:func.8">sup</a> <span class="p2">(<span class="default"><a href="supinf_2.html#K19" title="SUPINF_2:func.19">rng</a> <font color="Maroon" title="c6">L</font></span>)</span></span>)</span> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c19">p</font></span>)</span>
 
<b>by </b><i><a class="ref" href="supinf_2.html#T14" title="SUPINF_2:th.14">SUPINF_2:14</a>, <a class="txt" href="mesfunc5.html#E1:105_1_2_7_1_9_1_3"><i><font color="Green" title="E145">A264</font></i></a></i>;<br/>
<a NAME="E13:105_1_2_7_1_9_1_3"/><b>then </b><i><font color="Green" title="E150">A269</font></i>: 
 <a href="supinf_2.html#K8" title="SUPINF_2:func.8">sup</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K19" title="SUPINF_2:func.19">rng</a> <font color="Maroon" title="c6">L</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K8" title="SUPINF_2:func.8">sup</a> <span class="p2">(<span class="default"><a href="supinf_2.html#K19" title="SUPINF_2:func.19">rng</a> <font color="Maroon" title="c6">L</font></span>)</span></span>)</span> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c19">p</font></span>)</span>
 
<b>by </b><i><a class="ref" href="supinf_2.html#T18" title="SUPINF_2:th.18">SUPINF_2:18</a></i>;<br/>
<a NAME="E14:105_1_2_7_1_9_1_3"/>
 <a href="supinf_2.html#K8" title="SUPINF_2:func.8">sup</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K19" title="SUPINF_2:func.19">rng</a> <font color="Maroon" title="c6">L</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K8" title="SUPINF_2:func.8">sup</a> <span class="p2">(<span class="default"><a href="supinf_2.html#K19" title="SUPINF_2:func.19">rng</a> <font color="Maroon" title="c6">L</font></span>)</span></span>)</span> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c19">p</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="105_1_2_7_1_9_1_3_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:105_1_2_7_1_9_1_3_1"/><i><font color="Green" title="E151">A270</font></i>: 
 <a href="supinf_2.html#K8" title="SUPINF_2:func.8">sup</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K19" title="SUPINF_2:func.19">rng</a> <font color="Maroon" title="c6">L</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K8" title="SUPINF_2:func.8">sup</a> <span class="p2">(<span class="default"><a href="supinf_2.html#K19" title="SUPINF_2:func.19">rng</a> <font color="Maroon" title="c6">L</font></span>)</span></span>)</span> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c19">p</font></span>)</span>
 ;<br/>

<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c19">p</font></span>)</span> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <span class="p2">(<span class="default"><a href="supinf_2.html#K8" title="SUPINF_2:func.8">sup</a> <span class="p3">(<span class="default"><a href="supinf_2.html#K19" title="SUPINF_2:func.19">rng</a> <font color="Maroon" title="c6">L</font></span>)</span></span>)</span></span>)</span> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K3" title="SUPINF_2:func.3">-</a> <span class="p2">(<span class="default"><a href="supinf_2.html#K8" title="SUPINF_2:func.8">sup</a> <span class="p3">(<span class="default"><a href="supinf_2.html#K19" title="SUPINF_2:func.19">rng</a> <font color="Maroon" title="c6">L</font></span>)</span></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c19">p</font></span>)</span> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="supinf_2.html#K8" title="SUPINF_2:func.8">sup</a> <span class="p3">(<span class="default"><a href="supinf_2.html#K19" title="SUPINF_2:func.19">rng</a> <font color="Maroon" title="c6">L</font></span>)</span></span>)</span> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <span class="p2">(<span class="default"><a href="supinf_2.html#K3" title="SUPINF_2:func.3">-</a> <span class="p3">(<span class="default"><a href="supinf_2.html#K8" title="SUPINF_2:func.8">sup</a> <span class="p4">(<span class="default"><a href="supinf_2.html#K19" title="SUPINF_2:func.19">rng</a> <font color="Maroon" title="c6">L</font></span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="mesfunc5.html#E7:105_1_2_7_1_9_1"><i><font color="Green" title="E141">A258</font></i></a>, <a class="ref" href="extreal1.html#T8" title="EXTREAL1:th.8">EXTREAL1:8</a>, <a class="txt" href="mesfunc5.html#E11:105_1_2_7_1_9_1"><i><font color="Green" title="E143">A262</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c19">p</font></span>)</span> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> 0</span>)</span>
<b>by </b><i><a class="ref" href="extreal1.html#T9" title="EXTREAL1:th.9">EXTREAL1:9</a></i>
<br/>.= 
 <a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c19">p</font>
<b>by </b><i><a class="ref" href="supinf_2.html#T18" title="SUPINF_2:th.18">SUPINF_2:18</a></i>
;<br/>
<b>hence </b><a NAME="E3:105_1_2_7_1_9_1_3_1"/>
contradiction
 <b>by </b><i><a class="txt" href="mesfunc5.html#E1:105_1_2_7_1_9_1_3"><i><font color="Green" title="E145">A264</font></i></a>, <a class="txt" href="mesfunc5.html#E1:105_1_2_7_1_9_1_3_1"><i><font color="Green" title="E151">A270</font></i></a>, <a class="ref" href="extreal1.html#T9" title="EXTREAL1:th.9">EXTREAL1:9</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E15:105_1_2_7_1_9_1_3"/><b>then </b>
 <a href="supinf_2.html#K8" title="SUPINF_2:func.8">sup</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K19" title="SUPINF_2:func.19">rng</a> <font color="Maroon" title="c6">L</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K8" title="SUPINF_2:func.8">sup</a> <span class="p2">(<span class="default"><a href="supinf_2.html#K19" title="SUPINF_2:func.19">rng</a> <font color="Maroon" title="c6">L</font></span>)</span></span>)</span> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c19">p</font></span>)</span>
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E13:105_1_2_7_1_9_1_3"><i><font color="Green" title="E150">A269</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/>
<a NAME="E16:105_1_2_7_1_9_1_3"/><b>then </b>
<font color="Maroon" title="c6">L</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c23">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K8" title="SUPINF_2:func.8">sup</a> <span class="p2">(<span class="default"><a href="supinf_2.html#K19" title="SUPINF_2:func.19">rng</a> <font color="Maroon" title="c6">L</font></span>)</span></span>)</span> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c19">p</font></span>)</span>
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E11:105_1_2_7_1_9_1_3"><i><font color="Green" title="E149">A268</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E17:105_1_2_7_1_9_1_3"/><b>then </b><i><font color="Green" title="E151">A271</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">L</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c23">n</font></span>)</span> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K8" title="SUPINF_2:func.8">sup</a> <span class="p2">(<span class="default"><a href="supinf_2.html#K19" title="SUPINF_2:func.19">rng</a> <font color="Maroon" title="c6">L</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c19">p</font>
 
<b>by </b><i><a class="ref" href="extreal2.html#T30" title="EXTREAL2:th.30">EXTREAL2:30</a></i>;<br/>
<a NAME="E18:105_1_2_7_1_9_1_3"/>
 <a href="supinf_2.html#K8" title="SUPINF_2:func.8">sup</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K19" title="SUPINF_2:func.19">rng</a> <font color="Maroon" title="c6">L</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">L</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c23">n</font></span>)</span> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c19">p</font></span>)</span>
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E10:105_1_2_7_1_9_1_3"><i><font color="Green" title="E148">A267</font></i></a>, <a class="ref" href="extreal2.html#T29" title="EXTREAL2:th.29">EXTREAL2:29</a></i>;<br/>
<a NAME="E19:105_1_2_7_1_9_1_3"/><b>then </b>
<span class="p1">(<span class="default"><a href="supinf_2.html#K8" title="SUPINF_2:func.8">sup</a> <span class="p2">(<span class="default"><a href="supinf_2.html#K19" title="SUPINF_2:func.19">rng</a> <font color="Maroon" title="c6">L</font></span>)</span></span>)</span> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">L</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c23">n</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c19">p</font>
 
<b>by </b><i><a class="ref" href="extreal2.html#T30" title="EXTREAL2:th.30">EXTREAL2:30</a></i>;<br/>
<a NAME="E20:105_1_2_7_1_9_1_3"/><b>then </b>
 <a href="supinf_2.html#K3" title="SUPINF_2:func.3">-</a> <span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c19">p</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="supinf_2.html#K3" title="SUPINF_2:func.3">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="supinf_2.html#K8" title="SUPINF_2:func.8">sup</a> <span class="p3">(<span class="default"><a href="supinf_2.html#K19" title="SUPINF_2:func.19">rng</a> <font color="Maroon" title="c6">L</font></span>)</span></span>)</span> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">L</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c23">n</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="supinf_2.html#T17" title="SUPINF_2:th.17">SUPINF_2:17</a></i>;<br/>
<a NAME="E21:105_1_2_7_1_9_1_3"/><b>then </b>
 <a href="supinf_2.html#K3" title="SUPINF_2:func.3">-</a> <span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c19">p</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">L</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c23">n</font></span>)</span> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K8" title="SUPINF_2:func.8">sup</a> <span class="p2">(<span class="default"><a href="supinf_2.html#K19" title="SUPINF_2:func.19">rng</a> <font color="Maroon" title="c6">L</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="extreal2.html#T15" title="EXTREAL2:th.15">EXTREAL2:15</a></i>;<br/>
<b>hence </b><a NAME="E22:105_1_2_7_1_9_1_3"/>
<span class="p1"><a href="extreal1.html#K4" title="EXTREAL1:func.4">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">L</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c23">n</font></span>)</span> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <span class="p3">(<span class="default"><a href="supinf_2.html#K8" title="SUPINF_2:func.8">sup</a> <span class="p4">(<span class="default"><a href="supinf_2.html#K19" title="SUPINF_2:func.19">rng</a> <font color="Maroon" title="c6">L</font></span>)</span></span>)</span></span>)</span></span><a href="extreal1.html#K4" title="EXTREAL1:func.4">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c19">p</font>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E17:105_1_2_7_1_9_1_3"><i><font color="Green" title="E151">A271</font></i></a>, <a class="ref" href="extreal2.html#T59" title="EXTREAL2:th.59">EXTREAL2:59</a></i>;<br/>


</div><b>end;</b></div><b>reconsider </b><font color="Maroon" title="c19">h</font> =  <a href="supinf_2.html#K8" title="SUPINF_2:func.8">sup</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K19" title="SUPINF_2:func.19">rng</a> <font color="Maroon" title="c6">L</font></span>)</span> as  <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>  <b>by </b><i><a class="txt" href="mesfunc5.html#E7:105_1_2_7_1_9_1"><i><font color="Green" title="E141">A258</font></i></a>, <a class="txt" href="mesfunc5.html#E11:105_1_2_7_1_9_1"><i><font color="Green" title="E143">A262</font></i></a>, <a class="ref" href="extreal1.html#T1" title="EXTREAL1:th.1">EXTREAL1:1</a></i>;<br/><a NAME="E14:105_1_2_7_1_9_1"/><i><font color="Green" title="E145">A272</font></i>: 
 <a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c19">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_2.html#K8" title="SUPINF_2:func.8">sup</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K19" title="SUPINF_2:func.19">rng</a> <font color="Maroon" title="c6">L</font></span>)</span>
 ;<br/><a NAME="E15:105_1_2_7_1_9_1"/><b>then </b><i><font color="Green" title="E146">A273</font></i>: 
<font color="Maroon" title="c6">L</font> is <a href="mesfunc5.html#V7" title="MESFUNC5:attr.7">convergent_to_finite_number</a>
 <b>by </b><i><a class="ref" href="mesfunc5.html#D8" target="_self" title="MESFUNC5:def.8">Def8</a>, <a class="txt" href="mesfunc5.html#E12:105_1_2_7_1_9_1"><i><font color="Green" title="E144">A263</font></i></a></i>;<br/><b>hence </b><a NAME="E16:105_1_2_7_1_9_1"/>
<font color="Maroon" title="c6">L</font> is <a href="mesfunc5.html#V10" title="MESFUNC5:attr.10">convergent</a>
 <b>by </b><i><a class="ref" href="mesfunc5.html#D11" target="_self" title="MESFUNC5:def.11">Def11</a></i>;<br/><a NAME="E17:105_1_2_7_1_9_1"/><b>then </b><i><font color="Green" title="E147">A274</font></i>: 
 <a href="mesfunc5.html#K2" title="MESFUNC5:func.2">lim</a> <font color="Maroon" title="c6">L</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_2.html#K8" title="SUPINF_2:func.8">sup</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K19" title="SUPINF_2:func.19">rng</a> <font color="Maroon" title="c6">L</font></span>)</span>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E12:105_1_2_7_1_9_1"><i><font color="Green" title="E144">A263</font></i></a>, <a class="txt" href="mesfunc5.html#E14:105_1_2_7_1_9_1"><i><font color="Green" title="E145">A272</font></i></a>, <a class="txt" href="mesfunc5.html#E15:105_1_2_7_1_9_1"><i><font color="Green" title="E146">A273</font></i></a>, <a class="ref" href="mesfunc5.html#D12" target="_self" title="MESFUNC5:def.12">Def12</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="105_1_2_7_1_9_1_4"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c20">e</font> be  <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ;<br/><b>assume </b><a NAME="E1:105_1_2_7_1_9_1_4"/>
0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c20">e</font>
 ;<br/><b>then consider </b><font color="Maroon" title="c21">N0</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E3:105_1_2_7_1_9_1_4"/><i><font color="Green" title="E148">A275</font></i>: 
for <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Maroon" title="c21">N0</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">n</font> holds <br/><span class="p1">(<span class="default"><a href="mesfunc5.html#K5" title="MESFUNC5:func.5">integral'</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">g</font></span>)</span> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c20">e</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">L</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Olive" title="b1">n</font>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E18:105_1_2_7_1"><i><font color="Green" title="E136">A242</font></i></a></i>;<br/><a NAME="E4:105_1_2_7_1_9_1_4"/><i><font color="Green" title="E149">A276</font></i>: 
<span class="p1">(<span class="default"><a href="mesfunc5.html#K5" title="MESFUNC5:func.5">integral'</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">g</font></span>)</span> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c20">e</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">L</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c21">N0</font>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E3:105_1_2_7_1_9_1_4"><i><font color="Green" title="E148">A275</font></i></a></i>;<br/><a NAME="E5:105_1_2_7_1_9_1_4"/>
<font color="Maroon" title="c6">L</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c21">N0</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="supinf_2.html#K8" title="SUPINF_2:func.8">sup</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K19" title="SUPINF_2:func.19">rng</a> <font color="Maroon" title="c6">L</font></span>)</span>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E5:105_1_2_7_1_9_1"><i><font color="Green" title="E140">A256</font></i></a></i>;<br/><a NAME="E6:105_1_2_7_1_9_1_4"/><b>then </b>
<span class="p1">(<span class="default"><a href="mesfunc5.html#K5" title="MESFUNC5:func.5">integral'</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">g</font></span>)</span> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c20">e</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="supinf_2.html#K8" title="SUPINF_2:func.8">sup</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K19" title="SUPINF_2:func.19">rng</a> <font color="Maroon" title="c6">L</font></span>)</span>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E4:105_1_2_7_1_9_1_4"><i><font color="Green" title="E149">A276</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><b>hence </b><a NAME="E7:105_1_2_7_1_9_1_4"/>
 <a href="mesfunc5.html#K5" title="MESFUNC5:func.5">integral'</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">g</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="mesfunc5.html#K2" title="MESFUNC5:func.2">lim</a> <font color="Maroon" title="c6">L</font></span>)</span> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c20">e</font></span>)</span>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E17:105_1_2_7_1_9_1"><i><font color="Green" title="E147">A274</font></i></a>, <a class="ref" href="extreal2.html#T29" title="EXTREAL2:th.29">EXTREAL2:29</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E19:105_1_2_7_1_9_1"/>
 <a href="mesfunc5.html#K5" title="MESFUNC5:func.5">integral'</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">g</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="mesfunc5.html#K2" title="MESFUNC5:func.2">lim</a> <font color="Maroon" title="c6">L</font>
 <b>by </b><i><a class="ref" href="mesfunc5.html#T4" target="_self" title="MESFUNC5:th.4">Th4</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="105_1_2_7_1_9_2"><b>suppose </b></a><a NAME="E1:105_1_2_7_1_9_2"/><i><font color="Green" title="E139">A277</font></i>: 
for <font color="Olive" title="b1">K</font> being <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>   holds <br/>(  not 0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Olive" title="b1">K</font> or  ex <font color="Olive" title="b2">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st not <font color="Maroon" title="c6">L</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Olive" title="b2">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">K</font> )
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="105_1_2_7_1_9_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c18">K</font> be  <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ;<br/><b>assume </b><a NAME="E1:105_1_2_7_1_9_2_1"/>
0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c18">K</font>
 ;<br/><b>then consider </b><font color="Maroon" title="c19">N0</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E3:105_1_2_7_1_9_2_1"/><i><font color="Green" title="E140">A278</font></i>: 
 <a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c18">K</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">L</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c19">N0</font>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E1:105_1_2_7_1_9_2"><i><font color="Green" title="E139">A277</font></i></a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="105_1_2_7_1_9_2_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c20">n</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/><b>assume </b><a NAME="E1:105_1_2_7_1_9_2_1_1"/>
<font color="Maroon" title="c19">N0</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c20">n</font>
 ;<br/><a NAME="E2:105_1_2_7_1_9_2_1_1"/><b>then </b>
<font color="Maroon" title="c6">L</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c19">N0</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">L</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c20">n</font>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E19:105_1_2_7_1"><i><font color="Green" title="E137">A246</font></i></a></i>;<br/><b>hence </b><a NAME="E3:105_1_2_7_1_9_2_1_1"/>
 <a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c18">K</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">L</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c20">n</font>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E3:105_1_2_7_1_9_2_1"><i><font color="Green" title="E140">A278</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E5:105_1_2_7_1_9_2_1"/>
 ex <font color="Olive" title="b1">N0</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <br/>for <font color="Olive" title="b2">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">N0</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">n</font> holds <br/><font color="Maroon" title="c18">K</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">L</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Olive" title="b2">n</font>
 ;<br/></div><b>end;</b></div><a NAME="E3:105_1_2_7_1_9_2"/><b>then </b><i><font color="Green" title="E140">A279</font></i>: 
<font color="Maroon" title="c6">L</font> is <a href="mesfunc5.html#V8" title="MESFUNC5:attr.8">convergent_to_+infty</a>
 <b>by </b><i><a class="ref" href="mesfunc5.html#D9" target="_self" title="MESFUNC5:def.9">Def9</a></i>;<br/><b>hence </b><a NAME="E4:105_1_2_7_1_9_2"/>
<font color="Maroon" title="c6">L</font> is <a href="mesfunc5.html#V10" title="MESFUNC5:attr.10">convergent</a>
 <b>by </b><i><a class="ref" href="mesfunc5.html#D11" target="_self" title="MESFUNC5:def.11">Def11</a></i>;<br/><a NAME="E5:105_1_2_7_1_9_2"/><b>then </b>
 <a href="mesfunc5.html#K2" title="MESFUNC5:func.2">lim</a> <font color="Maroon" title="c6">L</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a> 
 <b>by </b><i><a class="txt" href="mesfunc5.html#E3:105_1_2_7_1_9_2"><i><font color="Green" title="E140">A279</font></i></a>, <a class="ref" href="mesfunc5.html#D12" target="_self" title="MESFUNC5:def.12">Def12</a></i>;<br/><b>hence </b><a NAME="E6:105_1_2_7_1_9_2"/>
 <a href="mesfunc5.html#K5" title="MESFUNC5:func.5">integral'</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">g</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="mesfunc5.html#K2" title="MESFUNC5:func.2">lim</a> <font color="Maroon" title="c6">L</font>
 <b>by </b><i><a class="ref" href="xxreal_0.html#T4" title="XXREAL_0:th.4">XXREAL_0:4</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="105_1_2_7_2"><b>suppose </b></a><a NAME="E1:105_1_2_7_2"/><i><font color="Green" title="E125">A280</font></i>: 
<font color="Maroon" title="c3">M</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">g</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a> 
 ;<br/><div class="add"><b>reconsider </b><font color="Maroon" title="c16">DG</font> =  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">g</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c2">S</font> <b>by </b><i><a class="txt" href="mesfunc5.html#E1:105"><i><font color="Green" title="E99">A1</font></i></a>, <a class="ref" href="mesfunc5.html#T43" target="_self" title="MESFUNC5:th.43">Th43</a></i>;<br/><a NAME="E3:105_1_2_7_2"/><i><font color="Green" title="E126">A281</font></i>: 
for <font color="Olive" title="b1">e</font> being  <a href="supinf_1.html#NM3" title="SUPINF_1:NM.3">R_eal</a>  st 0 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">e</font> &amp; <font color="Olive" title="b1">e</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <span class="p1">(<span class="default"><a href="seq_4.html#K3" title="SEQ_4:func.3">min</a> <font color="Maroon" title="c10">RINGA</font></span>)</span> holds <br/>for <font color="Olive" title="b2">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <span class="p3">(<span class="default"><a href="seq_4.html#K3" title="SEQ_4:func.3">min</a> <font color="Maroon" title="c10">RINGA</font></span>)</span></span>)</span> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <font color="Olive" title="b1">e</font></span>)</span> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <span class="p2">(<span class="default"><a href="mesfunc1.html#K15" title="MESFUNC1:func.15">less_dom</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="mesfunc1.html#K6" title="MESFUNC1:func.6">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Olive" title="b2">n</font></span>)</span></span>)</span>,<font color="Olive" title="b1">e</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="mesfunc5.html#K5" title="MESFUNC5:func.5">integral'</a> <font color="Maroon" title="c3">M</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Olive" title="b2">n</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="105_1_2_7_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c17">e</font> be   <a href="supinf_1.html#NM3" title="SUPINF_1:NM.3">R_eal</a>;<br/>

<b>assume </b><a NAME="E1:105_1_2_7_2_1"/><i><font color="Green" title="E127">A282</font></i>: 
( 0 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c17">e</font> &amp; <font color="Maroon" title="c17">e</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <span class="p1">(<span class="default"><a href="seq_4.html#K3" title="SEQ_4:func.3">min</a> <font color="Maroon" title="c10">RINGA</font></span>)</span> )
 ;<br/>

<a NAME="E2:105_1_2_7_2_1"/><b>then </b><i><font color="Green" title="E128">A283</font></i>: 
( <font color="Maroon" title="c17">e</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a>  &amp; <font color="Maroon" title="c17">e</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>  )
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T4" title="XXREAL_0:th.4">XXREAL_0:4</a></i>;<br/>
<a NAME="E3:105_1_2_7_2_1"/><b>then </b>
<font color="Maroon" title="c17">e</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a> 
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T4" title="XXREAL_0:th.4">XXREAL_0:4</a></i>;<br/>
<a NAME="E4:105_1_2_7_2_1"/><b>then </b><i><font color="Green" title="E129">A284</font></i>: 
0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <span class="p2">(<span class="default"><a href="seq_4.html#K3" title="SEQ_4:func.3">min</a> <font color="Maroon" title="c10">RINGA</font></span>)</span></span>)</span> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <font color="Maroon" title="c17">e</font>
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E1:105_1_2_7_2_1"><i><font color="Green" title="E127">A282</font></i></a>, <a class="ref" href="extreal1.html#T40" title="EXTREAL1:th.40">EXTREAL1:40</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c18">H</font> being   <a href="prob_1.html#NM2" title="PROB_1:NM.2">SetSequence</a> of <font color="Maroon" title="c1">X</font>, <font color="Maroon" title="c19">MF</font> being   <a href="mesfunc5.html#NM1" title="MESFUNC5:NM.1">ExtREAL_sequence</a><b> such that </b><br/><a NAME="E6:105_1_2_7_2_1"/><i><font color="Green" title="E130">A285</font></i>: 
for <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <font color="Maroon" title="c18">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mesfunc1.html#K15" title="MESFUNC1:func.15">less_dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="mesfunc1.html#K6" title="MESFUNC1:func.6">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Olive" title="b1">n</font></span>)</span></span>)</span>,<font color="Maroon" title="c17">e</font>
 <b>and </b><br/><a NAME="E7:105_1_2_7_2_1"/>
for <font color="Olive" title="b1">n</font>, <font color="Olive" title="b2">m</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">m</font> holds <br/><font color="Maroon" title="c18">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c18">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">m</font>
 <b>and </b><br/><a NAME="E8:105_1_2_7_2_1"/><i><font color="Green" title="E131">A286</font></i>: 
for <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <font color="Maroon" title="c18">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">g</font>
 <b>and </b><br/><a NAME="E9:105_1_2_7_2_1"/>
for <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <font color="Maroon" title="c19">MF</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">M</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font></span>)</span>
 <b>and </b><br/><a NAME="E10:105_1_2_7_2_1"/>
<font color="Maroon" title="c3">M</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">g</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_2.html#K8" title="SUPINF_2:func.8">sup</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K19" title="SUPINF_2:func.19">rng</a> <font color="Maroon" title="c19">MF</font></span>)</span>
 <b>and </b><br/><a NAME="E11:105_1_2_7_2_1"/><i><font color="Green" title="E132">A287</font></i>: 
for <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <font color="Maroon" title="c18">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">S</font>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E1:105_1_2_7_2_1"><i><font color="Green" title="E127">A282</font></i></a>, <a class="txt" href="mesfunc5.html#E31:105_1_2"><i><font color="Green" title="E121">A37</font></i></a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="105_1_2_7_2_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c20">n</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/><a NAME="E1:105_1_2_7_2_1_1"/>
<font color="Maroon" title="c18">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">n</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c16">DG</font>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E8:105_1_2_7_2_1"><i><font color="Green" title="E131">A286</font></i></a></i>;<br/><b>then </b><i><font color="Green" title="E133">A288</font></i>: <font color="Maroon" title="c16">DG</font> = 
<font color="Maroon" title="c16">DG</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">n</font></span>)</span>
<b>by </b><i><a class="ref" href="xboole_1.html#T12" title="XBOOLE_1:th.12">XBOOLE_1:12</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c16">DG</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p2">(<span class="default"><font color="Maroon" title="c18">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">n</font></span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">n</font></span>)</span>
<b>by </b><i><a class="ref" href="xboole_1.html#T39" title="XBOOLE_1:th.39">XBOOLE_1:39</a></i>
;<br/><a NAME="E3:105_1_2_7_2_1_1"/>
<font color="Maroon" title="c18">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">n</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c2">S</font>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E11:105_1_2_7_2_1"><i><font color="Green" title="E132">A287</font></i></a></i>;<br/><a NAME="E4:105_1_2_7_2_1_1"/><b>then </b>
<font color="Maroon" title="c18">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">n</font> <a href="measure1.html#R2" title="MEASURE1:pred.2">is_measurable</a> <font color="Maroon" title="c3">M</font>
 <b>by </b><i><a class="ref" href="measure1.html#D12" title="MEASURE1:def.12">MEASURE1:def 12</a></i>;<br/><a NAME="E5:105_1_2_7_2_1_1"/><b>then </b><i><font color="Green" title="E134">A289</font></i>: 
<font color="Maroon" title="c1">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">n</font></span>)</span> <a href="measure1.html#R2" title="MEASURE1:pred.2">is_measurable</a> <font color="Maroon" title="c3">M</font>
 <b>by </b><i><a class="ref" href="measure1.html#T66" title="MEASURE1:th.66">MEASURE1:66</a></i>;<br/><a NAME="E6:105_1_2_7_2_1_1"/>
<font color="Maroon" title="c16">DG</font> <a href="measure1.html#R2" title="MEASURE1:pred.2">is_measurable</a> <font color="Maroon" title="c3">M</font>
 <b>by </b><i><a class="ref" href="measure1.html#D12" title="MEASURE1:def.12">MEASURE1:def 12</a></i>;<br/><a NAME="E7:105_1_2_7_2_1_1"/><b>then </b><i><font color="Green" title="E135">A290</font></i>: 
<font color="Maroon" title="c16">DG</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2">(<span class="default"><font color="Maroon" title="c18">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">n</font></span>)</span></span>)</span> <a href="measure1.html#R2" title="MEASURE1:pred.2">is_measurable</a> <font color="Maroon" title="c3">M</font>
 <b>by </b><i><a class="ref" href="measure1.html#T66" title="MEASURE1:th.66">MEASURE1:66</a>, <a class="txt" href="mesfunc5.html#E5:105_1_2_7_2_1_1"><i><font color="Green" title="E134">A289</font></i></a></i>;<br/><font color="Maroon" title="c16">DG</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2">(<span class="default"><font color="Maroon" title="c18">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">n</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c16">DG</font> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">n</font></span>)</span>
<b>by </b><i><a class="ref" href="xboole_1.html#T49" title="XBOOLE_1:th.49">XBOOLE_1:49</a></i>
<br/>.= 
<font color="Maroon" title="c16">DG</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">n</font></span>)</span>
<b>by </b><i><a class="ref" href="xboole_1.html#T28" title="XBOOLE_1:th.28">XBOOLE_1:28</a></i>
;<br/><b>then reconsider </b><font color="Maroon" title="c21">A</font> = <font color="Maroon" title="c16">DG</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">n</font></span>)</span> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c2">S</font> <b>by </b><i><a class="txt" href="mesfunc5.html#E7:105_1_2_7_2_1_1"><i><font color="Green" title="E135">A290</font></i></a>, <a class="ref" href="measure1.html#D12" title="MEASURE1:def.12">MEASURE1:def 12</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c22">B</font> = <font color="Maroon" title="c18">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">n</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c2">S</font> <b>by </b><i><a class="txt" href="mesfunc5.html#E11:105_1_2_7_2_1"><i><font color="Green" title="E132">A287</font></i></a></i>;<br/><a NAME="E11:105_1_2_7_2_1_1"/><i><font color="Green" title="E136">A291</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c20">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">g</font>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E4:105"><i><font color="Green" title="E102">A4</font></i></a></i>;<br/><a NAME="E12:105_1_2_7_2_1_1"/><b>then </b><i><font color="Green" title="E137">A292</font></i>: 
<font color="Maroon" title="c22">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c20">n</font></span>)</span>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E2:105_1_2_7_2_1_1"><i><font color="Green" title="E133">A288</font></i></a>, <a class="ref" href="xboole_1.html#T7" title="XBOOLE_1:th.7">XBOOLE_1:7</a></i>;<br/><a NAME="E13:105_1_2_7_2_1_1"/><i><font color="Green" title="E138">A293</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c20">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">A</font> <a href="measure1.html#K2" title="MEASURE1:func.2">\/</a> <font color="Maroon" title="c22">B</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c20">n</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E11:105_1_2_7_2_1_1"><i><font color="Green" title="E136">A291</font></i></a>, <a class="txt" href="mesfunc5.html#E2:105_1_2_7_2_1_1"><i><font color="Green" title="E133">A288</font></i></a></i>;<br/><a NAME="E14:105_1_2_7_2_1_1"/>
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c20">n</font></span>)</span> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c20">n</font></span>)</span> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c20">n</font></span>)</span> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Olive" title="b1">x</font>
 ;<br/><a NAME="E15:105_1_2_7_2_1_1"/><b>then </b><i><font color="Green" title="E139">A294</font></i>: 
<font color="Maroon" title="c5">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c20">n</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c20">n</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">A</font> <a href="measure1.html#K2" title="MEASURE1:func.2">\/</a> <font color="Maroon" title="c22">B</font></span>)</span>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E13:105_1_2_7_2_1_1"><i><font color="Green" title="E138">A293</font></i></a>, <a class="ref" href="funct_1.html#T68" title="FUNCT_1:th.68">FUNCT_1:68</a></i>;<br/><a NAME="E16:105_1_2_7_2_1_1"/><i><font color="Green" title="E140">A295</font></i>: 
<font color="Maroon" title="c5">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c20">n</font> is <a href="supinf_2.html#V6" title="SUPINF_2:attr.6">nonnegative</a>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E5:105"><i><font color="Green" title="E103">A5</font></i></a></i>;<br/><a NAME="E17:105_1_2_7_2_1_1"/><i><font color="Green" title="E141">A296</font></i>: 
<font color="Maroon" title="c21">A</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c22">B</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T79" title="XBOOLE_1:th.79">XBOOLE_1:79</a></i>;<br/><a NAME="E18:105_1_2_7_2_1_1"/><i><font color="Green" title="E142">A297</font></i>: 
<font color="Maroon" title="c5">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c20">n</font> <a href="mesfunc2.html#R1" title="MESFUNC2:pred.1">is_simple_func_in</a> <font color="Maroon" title="c2">S</font>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E3:105"><i><font color="Green" title="E101">A3</font></i></a></i>;<br/><a NAME="E19:105_1_2_7_2_1_1"/><b>then </b><i><font color="Green" title="E143">A298</font></i>: 
 <a href="mesfunc5.html#K5" title="MESFUNC5:func.5">integral'</a> <font color="Maroon" title="c3">M</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c20">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="mesfunc5.html#K5" title="MESFUNC5:func.5">integral'</a> <font color="Maroon" title="c3">M</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c20">n</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c21">A</font></span>)</span></span>)</span> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <span class="p1">(<span class="default"><a href="mesfunc5.html#K5" title="MESFUNC5:func.5">integral'</a> <font color="Maroon" title="c3">M</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c20">n</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c22">B</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E16:105_1_2_7_2_1_1"><i><font color="Green" title="E140">A295</font></i></a>, <a class="txt" href="mesfunc5.html#E17:105_1_2_7_2_1_1"><i><font color="Green" title="E141">A296</font></i></a>, <a class="ref" href="mesfunc5.html#T73" target="_self" title="MESFUNC5:th.73">Th73</a>, <a class="txt" href="mesfunc5.html#E15:105_1_2_7_2_1_1"><i><font color="Green" title="E139">A294</font></i></a></i>;<br/><a NAME="E20:105_1_2_7_2_1_1"/><i><font color="Green" title="E144">A299</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c20">n</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c21">A</font> is <a href="supinf_2.html#V6" title="SUPINF_2:attr.6">nonnegative</a> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c20">n</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c22">B</font> is <a href="supinf_2.html#V6" title="SUPINF_2:attr.6">nonnegative</a> )
 <b>by </b><i><a class="txt" href="mesfunc5.html#E16:105_1_2_7_2_1_1"><i><font color="Green" title="E140">A295</font></i></a>, <a class="ref" href="mesfunc5.html#T21" target="_self" title="MESFUNC5:th.21">Th21</a></i>;<br/><a NAME="E21:105_1_2_7_2_1_1"/><i><font color="Green" title="E145">A300</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c20">n</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c21">A</font> <a href="mesfunc2.html#R1" title="MESFUNC2:pred.1">is_simple_func_in</a> <font color="Maroon" title="c2">S</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c20">n</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c22">B</font> <a href="mesfunc2.html#R1" title="MESFUNC2:pred.1">is_simple_func_in</a> <font color="Maroon" title="c2">S</font> )
 <b>by </b><i><a class="txt" href="mesfunc5.html#E18:105_1_2_7_2_1_1"><i><font color="Green" title="E142">A297</font></i></a>, <a class="ref" href="mesfunc5.html#T40" target="_self" title="MESFUNC5:th.40">Th40</a></i>;<br/><a NAME="E22:105_1_2_7_2_1_1"/><b>then </b>
( 0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="mesfunc5.html#K5" title="MESFUNC5:func.5">integral'</a> <font color="Maroon" title="c3">M</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c20">n</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c21">A</font></span>)</span> &amp; 0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="mesfunc5.html#K5" title="MESFUNC5:func.5">integral'</a> <font color="Maroon" title="c3">M</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c20">n</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c22">B</font></span>)</span> )
 <b>by </b><i><a class="ref" href="mesfunc5.html#T74" target="_self" title="MESFUNC5:th.74">Th74</a>, <a class="txt" href="mesfunc5.html#E20:105_1_2_7_2_1_1"><i><font color="Green" title="E144">A299</font></i></a></i>;<br/><a NAME="E23:105_1_2_7_2_1_1"/><b>then </b><i><font color="Green" title="E146">A301</font></i>: 
 <a href="mesfunc5.html#K5" title="MESFUNC5:func.5">integral'</a> <font color="Maroon" title="c3">M</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c20">n</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c22">B</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="mesfunc5.html#K5" title="MESFUNC5:func.5">integral'</a> <font color="Maroon" title="c3">M</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c20">n</font></span>)</span>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E19:105_1_2_7_2_1_1"><i><font color="Green" title="E143">A298</font></i></a>, <a class="ref" href="supinf_2.html#T20" title="SUPINF_2:th.20">SUPINF_2:20</a></i>;<br/><i><font color="Green" title="E147">A302</font></i>:  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c20">n</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c22">B</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c20">n</font></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c22">B</font>
<b>by </b><i><a class="ref" href="funct_1.html#T68" title="FUNCT_1:th.68">FUNCT_1:68</a></i>
<br/>.= 
<font color="Maroon" title="c22">B</font>
<b>by </b><i><a class="txt" href="mesfunc5.html#E12:105_1_2_7_2_1_1"><i><font color="Green" title="E137">A292</font></i></a>, <a class="ref" href="xboole_1.html#T28" title="XBOOLE_1:th.28">XBOOLE_1:28</a></i>
;<br/><a NAME="E25:105_1_2_7_2_1_1"/>
<span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <span class="p2">(<span class="default"><a href="seq_4.html#K3" title="SEQ_4:func.3">min</a> <font color="Maroon" title="c10">RINGA</font></span>)</span></span>)</span> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <font color="Maroon" title="c17">e</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a> 
 <b>by </b><i><a class="txt" href="mesfunc5.html#E1:105_1_2_7_2_1"><i><font color="Green" title="E127">A282</font></i></a>, <a class="ref" href="supinf_2.html#T10" title="SUPINF_2:th.10">SUPINF_2:10</a></i>;<br/><b>then consider </b><font color="Maroon" title="c23">EP</font> being   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <font color="Maroon" title="c1">X</font>, <a href="numbers.html#K6" title="NUMBERS:func.6">ExtREAL</a> <b> such that </b><br/><a NAME="E27:105_1_2_7_2_1_1"/><i><font color="Green" title="E148">A303</font></i>: 
( <font color="Maroon" title="c23">EP</font> <a href="mesfunc2.html#R1" title="MESFUNC2:pred.1">is_simple_func_in</a> <font color="Maroon" title="c2">S</font> &amp;  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c23">EP</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">B</font> &amp; ( for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c22">B</font> holds <br/><font color="Maroon" title="c23">EP</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <span class="p2">(<span class="default"><a href="seq_4.html#K3" title="SEQ_4:func.3">min</a> <font color="Maroon" title="c10">RINGA</font></span>)</span></span>)</span> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <font color="Maroon" title="c17">e</font> ) )
 <b>by </b><i><a class="ref" href="mesfunc5.html#T47" target="_self" title="MESFUNC5:th.47">Th47</a>, <a class="txt" href="mesfunc5.html#E4:105_1_2_7_2_1"><i><font color="Green" title="E129">A284</font></i></a></i>;<br/><a NAME="E28:105_1_2_7_2_1_1"/>
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c23">EP</font> holds <br/>0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c23">EP</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Olive" title="b1">x</font>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E27:105_1_2_7_2_1_1"><i><font color="Green" title="E148">A303</font></i></a>, <a class="txt" href="mesfunc5.html#E4:105_1_2_7_2_1"><i><font color="Green" title="E129">A284</font></i></a></i>;<br/><a NAME="E29:105_1_2_7_2_1_1"/><b>then </b><i><font color="Green" title="E149">A304</font></i>: 
<font color="Maroon" title="c23">EP</font> is <a href="supinf_2.html#V6" title="SUPINF_2:attr.6">nonnegative</a>
 <b>by </b><i><a class="ref" href="supinf_2.html#T71" title="SUPINF_2:th.71">SUPINF_2:71</a></i>;<br/><a NAME="E30:105_1_2_7_2_1_1"/><i><font color="Green" title="E150">A305</font></i>: 
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c20">n</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c22">B</font></span>)</span> <a href="mesfunc1.html#K6" title="MESFUNC1:func.6">-</a> <font color="Maroon" title="c23">EP</font></span>)</span> holds <br/><font color="Maroon" title="c23">EP</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Olive" title="b1">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c20">n</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c22">B</font></span>)</span> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Olive" title="b1">x</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="105_1_2_7_2_1_1_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c24">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:105_1_2_7_2_1_1_4"/>
<font color="Maroon" title="c24">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c20">n</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c22">B</font></span>)</span> <a href="mesfunc1.html#K6" title="MESFUNC1:func.6">-</a> <font color="Maroon" title="c23">EP</font></span>)</span>
 ;<br/>

<a NAME="E2:105_1_2_7_2_1_1_4"/><b>then </b>
<font color="Maroon" title="c24">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c20">n</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c22">B</font></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c23">EP</font></span>)</span></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c20">n</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c22">B</font></span>)</span> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p4"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p3">(<span class="default"><font color="Maroon" title="c23">EP</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p4"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c20">n</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c22">B</font></span>)</span> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p4"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p3">(<span class="default"><font color="Maroon" title="c23">EP</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p4"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="mesfunc1.html#D4" title="MESFUNC1:def.4">MESFUNC1:def 4</a></i>;<br/>
<a NAME="E3:105_1_2_7_2_1_1_4"/><b>then </b><i><font color="Green" title="E151">A306</font></i>: 
<font color="Maroon" title="c24">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c20">n</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c22">B</font></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c23">EP</font></span>)</span>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/>
<a NAME="E4:105_1_2_7_2_1_1_4"/><b>then </b><i><font color="Green" title="E152">A307</font></i>: 
<font color="Maroon" title="c24">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c20">n</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c22">B</font></span>)</span>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E5:105_1_2_7_2_1_1_4"/><b>then </b><i><font color="Green" title="E153">A308</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c20">n</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c22">B</font></span>)</span> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c24">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c20">n</font></span>)</span> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c24">x</font>
 
<b>by </b><i><a class="ref" href="funct_1.html#T68" title="FUNCT_1:th.68">FUNCT_1:68</a></i>;<br/>
<b>set </b><font color="Maroon" title="c25">f</font> = <font color="Maroon" title="c4">g</font> <a href="mesfunc1.html#K6" title="MESFUNC1:func.6">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c20">n</font></span>)</span>;<br/>
<a NAME="E6:105_1_2_7_2_1_1_4"/><i><font color="Green" title="E154">A309</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="mesfunc1.html#K6" title="MESFUNC1:func.6">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c20">n</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">g</font>
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E30:105_1_2"><i><font color="Green" title="E120">A34</font></i></a></i>;<br/>
<a NAME="E7:105_1_2_7_2_1_1_4"/>
<font color="Maroon" title="c24">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="mesfunc1.html#K15" title="MESFUNC1:func.15">less_dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="mesfunc1.html#K6" title="MESFUNC1:func.6">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c20">n</font></span>)</span></span>)</span>,<font color="Maroon" title="c17">e</font>
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E6:105_1_2_7_2_1"><i><font color="Green" title="E130">A285</font></i></a>, <a class="txt" href="mesfunc5.html#E4:105_1_2_7_2_1_1_4"><i><font color="Green" title="E152">A307</font></i></a>, <a class="txt" href="mesfunc5.html#E24:105_1_2_7_2_1_1"><i><font color="Green" title="E147">A302</font></i></a></i>;<br/>
<a NAME="E8:105_1_2_7_2_1_1_4"/><b>then </b><i><font color="Green" title="E155">A310</font></i>: 
( <font color="Maroon" title="c24">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="mesfunc1.html#K6" title="MESFUNC1:func.6">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c20">n</font></span>)</span></span>)</span> &amp;  ex <font color="Olive" title="b1">y</font> being  <a href="supinf_1.html#NM3" title="SUPINF_1:NM.3">R_eal</a> st <br/>( <font color="Olive" title="b1">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="mesfunc1.html#K6" title="MESFUNC1:func.6">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c20">n</font></span>)</span></span>)</span> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c24">x</font> &amp; <font color="Olive" title="b1">y</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c17">e</font> ) )
 
<b>by </b><i><a class="ref" href="mesfunc1.html#D12" title="MESFUNC1:def.12">MESFUNC1:def 12</a></i>;<br/>
<a NAME="E9:105_1_2_7_2_1_1_4"/><b>then </b><i><font color="Green" title="E156">A311</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c24">x</font></span>)</span> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c20">n</font></span>)</span> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c24">x</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c17">e</font>
 
<b>by </b><i><a class="ref" href="mesfunc1.html#D4" title="MESFUNC1:def.4">MESFUNC1:def 4</a></i>;<br/>
<a NAME="E10:105_1_2_7_2_1_1_4"/>
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">g</font> holds <br/>( <font color="Maroon" title="c4">g</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a>  &amp; <font color="Maroon" title="c4">g</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>  )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="105_1_2_7_2_1_1_4_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c26">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:105_1_2_7_2_1_1_4_1"/>
<font color="Maroon" title="c26">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">g</font>
 ;<br/>

<a NAME="E2:105_1_2_7_2_1_1_4_1"/><b>then </b>
<span class="p1"><a href="extreal1.html#K4" title="EXTREAL1:func.4">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c26">x</font></span>)</span></span><a href="extreal1.html#K4" title="EXTREAL1:func.4">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a> 
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E29:105_1_2"><i><font color="Green" title="E119">A33</font></i></a>, <a class="ref" href="mesfunc2.html#D1" title="MESFUNC2:def.1">MESFUNC2:def 1</a></i>;<br/>
<b>hence </b><a NAME="E3:105_1_2_7_2_1_1_4_1"/>
( <font color="Maroon" title="c4">g</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c26">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a>  &amp; <font color="Maroon" title="c4">g</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c26">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>  )
 <b>by </b><i><a class="ref" href="extreal2.html#T67" title="EXTREAL2:th.67">EXTREAL2:67</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E11:105_1_2_7_2_1_1_4"/><b>then </b>
( <font color="Maroon" title="c4">g</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c24">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a>  &amp; <font color="Maroon" title="c4">g</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c24">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>  )
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E8:105_1_2_7_2_1_1_4"><i><font color="Green" title="E155">A310</font></i></a>, <a class="txt" href="mesfunc5.html#E6:105_1_2_7_2_1_1_4"><i><font color="Green" title="E154">A309</font></i></a></i>;<br/>
<a NAME="E12:105_1_2_7_2_1_1_4"/><b>then </b>
<font color="Maroon" title="c4">g</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c24">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c20">n</font></span>)</span> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c24">x</font></span>)</span> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <font color="Maroon" title="c17">e</font>
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E9:105_1_2_7_2_1_1_4"><i><font color="Green" title="E156">A311</font></i></a>, <a class="ref" href="extreal2.html#T33" title="EXTREAL2:th.33">EXTREAL2:33</a>, <a class="txt" href="mesfunc5.html#E2:105_1_2_7_2_1"><i><font color="Green" title="E128">A283</font></i></a></i>;<br/>
<a NAME="E13:105_1_2_7_2_1_1_4"/><b>then </b><i><font color="Green" title="E157">A312</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c24">x</font></span>)</span> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <font color="Maroon" title="c17">e</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c20">n</font></span>)</span> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c24">x</font>
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E2:105_1_2_7_2_1"><i><font color="Green" title="E128">A283</font></i></a>, <a class="ref" href="extreal2.html#T34" title="EXTREAL2:th.34">EXTREAL2:34</a></i>;<br/>
<a NAME="E14:105_1_2_7_2_1_1_4"/>
 <a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <span class="p1">(<span class="default"><a href="seq_4.html#K3" title="SEQ_4:func.3">min</a> <font color="Maroon" title="c10">RINGA</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">g</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c24">x</font>
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E34:105_1_2"><i><font color="Green" title="E124">A117</font></i></a>, <a class="txt" href="mesfunc5.html#E6:105_1_2_7_2_1_1_4"><i><font color="Green" title="E154">A309</font></i></a>, <a class="txt" href="mesfunc5.html#E8:105_1_2_7_2_1_1_4"><i><font color="Green" title="E155">A310</font></i></a></i>;<br/>
<a NAME="E15:105_1_2_7_2_1_1_4"/><b>then </b>
<span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <span class="p2">(<span class="default"><a href="seq_4.html#K3" title="SEQ_4:func.3">min</a> <font color="Maroon" title="c10">RINGA</font></span>)</span></span>)</span> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <font color="Maroon" title="c17">e</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c24">x</font></span>)</span> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <font color="Maroon" title="c17">e</font>
 
<b>by </b><i><a class="ref" href="supinf_2.html#T15" title="SUPINF_2:th.15">SUPINF_2:15</a>, <a class="txt" href="mesfunc5.html#E2:105_1_2_7_2_1"><i><font color="Green" title="E128">A283</font></i></a></i>;<br/>
<a NAME="E16:105_1_2_7_2_1_1_4"/><b>then </b>
<span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <span class="p2">(<span class="default"><a href="seq_4.html#K3" title="SEQ_4:func.3">min</a> <font color="Maroon" title="c10">RINGA</font></span>)</span></span>)</span> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <font color="Maroon" title="c17">e</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c20">n</font></span>)</span> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c24">x</font>
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E13:105_1_2_7_2_1_1_4"><i><font color="Green" title="E157">A312</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<b>hence </b><a NAME="E17:105_1_2_7_2_1_1_4"/>
<font color="Maroon" title="c23">EP</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c24">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c20">n</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c22">B</font></span>)</span> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c24">x</font>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E5:105_1_2_7_2_1_1_4"><i><font color="Green" title="E153">A308</font></i></a>, <a class="txt" href="mesfunc5.html#E3:105_1_2_7_2_1_1_4"><i><font color="Green" title="E151">A306</font></i></a>, <a class="txt" href="mesfunc5.html#E24:105_1_2_7_2_1_1"><i><font color="Green" title="E147">A302</font></i></a>, <a class="txt" href="mesfunc5.html#E27:105_1_2_7_2_1_1"><i><font color="Green" title="E148">A303</font></i></a></i>;<br/>


</div><b>end;</b></div><a NAME="E31:105_1_2_7_2_1_1"/><b>then </b><i><font color="Green" title="E151">A313</font></i>: 
 <a href="mesfunc5.html#K5" title="MESFUNC5:func.5">integral'</a> <font color="Maroon" title="c3">M</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c23">EP</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c20">n</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c22">B</font></span>)</span> <a href="mesfunc1.html#K6" title="MESFUNC1:func.6">-</a> <font color="Maroon" title="c23">EP</font></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="mesfunc5.html#K5" title="MESFUNC5:func.5">integral'</a> <font color="Maroon" title="c3">M</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c20">n</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c22">B</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c20">n</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c22">B</font></span>)</span> <a href="mesfunc1.html#K6" title="MESFUNC1:func.6">-</a> <font color="Maroon" title="c23">EP</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="mesfunc5.html#T76" target="_self" title="MESFUNC5:th.76">Th76</a>, <a class="txt" href="mesfunc5.html#E21:105_1_2_7_2_1_1"><i><font color="Green" title="E145">A300</font></i></a>, <a class="txt" href="mesfunc5.html#E27:105_1_2_7_2_1_1"><i><font color="Green" title="E148">A303</font></i></a>, <a class="txt" href="mesfunc5.html#E20:105_1_2_7_2_1_1"><i><font color="Green" title="E144">A299</font></i></a>, <a class="txt" href="mesfunc5.html#E29:105_1_2_7_2_1_1"><i><font color="Green" title="E149">A304</font></i></a></i>;<br/><a NAME="E32:105_1_2_7_2_1_1"/><i><font color="Green" title="E152">A314</font></i>: 
 <a href="mesfunc5.html#K5" title="MESFUNC5:func.5">integral'</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c23">EP</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <span class="p3">(<span class="default"><a href="seq_4.html#K3" title="SEQ_4:func.3">min</a> <font color="Maroon" title="c10">RINGA</font></span>)</span></span>)</span> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <font color="Maroon" title="c17">e</font></span>)</span> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c22">B</font></span>)</span>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E27:105_1_2_7_2_1_1"><i><font color="Green" title="E148">A303</font></i></a>, <a class="txt" href="mesfunc5.html#E4:105_1_2_7_2_1"><i><font color="Green" title="E129">A284</font></i></a>, <a class="ref" href="mesfunc5.html#T77" target="_self" title="MESFUNC5:th.77">Th77</a></i>;<br/><a NAME="E33:105_1_2_7_2_1_1"/>
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c20">n</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c22">B</font></span>)</span> <a href="mesfunc1.html#K6" title="MESFUNC1:func.6">-</a> <font color="Maroon" title="c23">EP</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c20">n</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c22">B</font></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c23">EP</font></span>)</span>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E21:105_1_2_7_2_1_1"><i><font color="Green" title="E145">A300</font></i></a>, <a class="txt" href="mesfunc5.html#E27:105_1_2_7_2_1_1"><i><font color="Green" title="E148">A303</font></i></a>, <a class="txt" href="mesfunc5.html#E20:105_1_2_7_2_1_1"><i><font color="Green" title="E144">A299</font></i></a>, <a class="txt" href="mesfunc5.html#E29:105_1_2_7_2_1_1"><i><font color="Green" title="E149">A304</font></i></a>, <a class="txt" href="mesfunc5.html#E30:105_1_2_7_2_1_1"><i><font color="Green" title="E150">A305</font></i></a>, <a class="ref" href="mesfunc5.html#T75" target="_self" title="MESFUNC5:th.75">Th75</a></i>;<br/><a NAME="E34:105_1_2_7_2_1_1"/><b>then </b>
( <font color="Maroon" title="c23">EP</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c20">n</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c22">B</font></span>)</span> <a href="mesfunc1.html#K6" title="MESFUNC1:func.6">-</a> <font color="Maroon" title="c23">EP</font></span>)</span></span>)</span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c23">EP</font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c20">n</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c22">B</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c20">n</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c22">B</font></span>)</span> <a href="mesfunc1.html#K6" title="MESFUNC1:func.6">-</a> <font color="Maroon" title="c23">EP</font></span>)</span></span>)</span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c20">n</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c22">B</font> )
 <b>by </b><i><a class="ref" href="grfunc_1.html#T64" title="GRFUNC_1:th.64">GRFUNC_1:64</a>, <a class="txt" href="mesfunc5.html#E27:105_1_2_7_2_1_1"><i><font color="Green" title="E148">A303</font></i></a>, <a class="txt" href="mesfunc5.html#E24:105_1_2_7_2_1_1"><i><font color="Green" title="E147">A302</font></i></a></i>;<br/><a NAME="E35:105_1_2_7_2_1_1"/><b>then </b>
 <a href="mesfunc5.html#K5" title="MESFUNC5:func.5">integral'</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c23">EP</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="mesfunc5.html#K5" title="MESFUNC5:func.5">integral'</a> <font color="Maroon" title="c3">M</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c20">n</font></span>)</span>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E23:105_1_2_7_2_1_1"><i><font color="Green" title="E146">A301</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a>, <a class="txt" href="mesfunc5.html#E31:105_1_2_7_2_1_1"><i><font color="Green" title="E151">A313</font></i></a></i>;<br/><b>hence </b><a NAME="E36:105_1_2_7_2_1_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <span class="p3">(<span class="default"><a href="seq_4.html#K3" title="SEQ_4:func.3">min</a> <font color="Maroon" title="c10">RINGA</font></span>)</span></span>)</span> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <font color="Maroon" title="c17">e</font></span>)</span> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <span class="p2">(<span class="default"><a href="mesfunc1.html#K15" title="MESFUNC1:func.15">less_dom</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="mesfunc1.html#K6" title="MESFUNC1:func.6">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c20">n</font></span>)</span></span>)</span>,<font color="Maroon" title="c17">e</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="mesfunc5.html#K5" title="MESFUNC5:func.5">integral'</a> <font color="Maroon" title="c3">M</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c20">n</font></span>)</span>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E6:105_1_2_7_2_1"><i><font color="Green" title="E130">A285</font></i></a>, <a class="txt" href="mesfunc5.html#E32:105_1_2_7_2_1_1"><i><font color="Green" title="E152">A314</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E13:105_1_2_7_2_1"/>
for <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <span class="p3">(<span class="default"><a href="seq_4.html#K3" title="SEQ_4:func.3">min</a> <font color="Maroon" title="c10">RINGA</font></span>)</span></span>)</span> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <font color="Maroon" title="c17">e</font></span>)</span> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <span class="p2">(<span class="default"><a href="mesfunc1.html#K15" title="MESFUNC1:func.15">less_dom</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="mesfunc1.html#K6" title="MESFUNC1:func.6">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Olive" title="b1">n</font></span>)</span></span>)</span>,<font color="Maroon" title="c17">e</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="mesfunc5.html#K5" title="MESFUNC5:func.5">integral'</a> <font color="Maroon" title="c3">M</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Olive" title="b1">n</font></span>)</span>
 ;<br/>


</div><b>end;</b></div><a NAME="E4:105_1_2_7_2"/>
for <font color="Olive" title="b1">y</font> being <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>   st 0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Olive" title="b1">y</font> holds <br/> ex <font color="Olive" title="b2">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <br/>for <font color="Olive" title="b3">m</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b2">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b3">m</font> holds <br/><font color="Olive" title="b1">y</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">L</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Olive" title="b3">m</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="105_1_2_7_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c17">y</font> be  <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ;<br/>

<b>assume </b><a NAME="E1:105_1_2_7_2_2"/>
0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c17">y</font>
 ;<br/>

<b>set </b><font color="Maroon" title="c18">e</font> = <span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <span class="p2">(<span class="default"><a href="seq_4.html#K3" title="SEQ_4:func.3">min</a> <font color="Maroon" title="c10">RINGA</font></span>)</span></span>)</span> <a href="extreal1.html#K3" title="EXTREAL1:func.3">/</a> <span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> 2</span>)</span>;<br/>
<b>reconsider </b><font color="Maroon" title="c19">ralpha</font> =  <a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <span class="p1">(<span class="default"><a href="seq_4.html#K3" title="SEQ_4:func.3">min</a> <font color="Maroon" title="c10">RINGA</font></span>)</span> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> <b>by </b><i><a class="ref" href="xxreal_0.html#T14" title="XXREAL_0:th.14">XXREAL_0:14</a></i>;<br/>
<b>set </b><font color="Maroon" title="c20">a2</font> = <font color="Maroon" title="c19">ralpha</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2;<br/>
<a NAME="E3:105_1_2_7_2_2"/><i><font color="Green" title="E127">A315</font></i>: 
<span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <span class="p2">(<span class="default"><a href="seq_4.html#K3" title="SEQ_4:func.3">min</a> <font color="Maroon" title="c10">RINGA</font></span>)</span></span>)</span> <a href="extreal1.html#K3" title="EXTREAL1:func.3">/</a> <span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">ralpha</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2
 
<b>by </b><i><a class="ref" href="extreal1.html#T39" title="EXTREAL1:th.39">EXTREAL1:39</a></i>;<br/>
<a NAME="E4:105_1_2_7_2_2"/><b>then </b><i><font color="Green" title="E128">A316</font></i>: 
0 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <span class="p2">(<span class="default"><a href="seq_4.html#K3" title="SEQ_4:func.3">min</a> <font color="Maroon" title="c10">RINGA</font></span>)</span></span>)</span> <a href="extreal1.html#K3" title="EXTREAL1:func.3">/</a> <span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> 2</span>)</span>
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E33:105_1_2"><i><font color="Green" title="E123">A116</font></i></a>, <a class="ref" href="xreal_1.html#T217" title="XREAL_1:th.217">XREAL_1:217</a></i>;<br/>
<a NAME="E5:105_1_2_7_2_2"/><i><font color="Green" title="E129">A317</font></i>: 
<span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <span class="p2">(<span class="default"><a href="seq_4.html#K3" title="SEQ_4:func.3">min</a> <font color="Maroon" title="c10">RINGA</font></span>)</span></span>)</span> <a href="extreal1.html#K3" title="EXTREAL1:func.3">/</a> <span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> 2</span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <span class="p1">(<span class="default"><a href="seq_4.html#K3" title="SEQ_4:func.3">min</a> <font color="Maroon" title="c10">RINGA</font></span>)</span>
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E33:105_1_2"><i><font color="Green" title="E123">A116</font></i></a>, <a class="ref" href="xreal_1.html#T218" title="XREAL_1:th.218">XREAL_1:218</a>, <a class="txt" href="mesfunc5.html#E3:105_1_2_7_2_2"><i><font color="Green" title="E127">A315</font></i></a></i>;<br/>
<a NAME="E6:105_1_2_7_2_2"/><i><font color="Green" title="E130">A318</font></i>: 
<span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <span class="p2">(<span class="default"><a href="seq_4.html#K3" title="SEQ_4:func.3">min</a> <font color="Maroon" title="c10">RINGA</font></span>)</span></span>)</span> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <span class="p3">(<span class="default"><a href="seq_4.html#K3" title="SEQ_4:func.3">min</a> <font color="Maroon" title="c10">RINGA</font></span>)</span></span>)</span> <a href="extreal1.html#K3" title="EXTREAL1:func.3">/</a> <span class="p2">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> 2</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">ralpha</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">ralpha</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span>
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E3:105_1_2_7_2_2"><i><font color="Green" title="E127">A315</font></i></a>, <a class="ref" href="supinf_2.html#T5" title="SUPINF_2:th.5">SUPINF_2:5</a></i>;<br/>
<a NAME="E7:105_1_2_7_2_2"/><i><font color="Green" title="E131">A319</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c19">ralpha</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">ralpha</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c19">ralpha</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">ralpha</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span>
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E33:105_1_2"><i><font color="Green" title="E123">A116</font></i></a>, <a class="ref" href="xreal_1.html#T16" title="XREAL_1:th.16">XREAL_1:16</a></i>;<br/>
<a NAME="E8:105_1_2_7_2_2"/><b>then </b><i><font color="Green" title="E132">A320</font></i>: 
0 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <span class="p2">(<span class="default"><a href="seq_4.html#K3" title="SEQ_4:func.3">min</a> <font color="Maroon" title="c10">RINGA</font></span>)</span></span>)</span> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <span class="p3">(<span class="default"><a href="seq_4.html#K3" title="SEQ_4:func.3">min</a> <font color="Maroon" title="c10">RINGA</font></span>)</span></span>)</span> <a href="extreal1.html#K3" title="EXTREAL1:func.3">/</a> <span class="p2">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> 2</span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E3:105_1_2_7_2_2"><i><font color="Green" title="E127">A315</font></i></a>, <a class="ref" href="supinf_2.html#T5" title="SUPINF_2:th.5">SUPINF_2:5</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c21">y1</font> = <font color="Maroon" title="c17">y</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> <b>by </b><i><a class="ref" href="xreal_0.html#D1" title="XREAL_0:def.1">XREAL_0:def 1</a></i>;<br/>
<a NAME="E10:105_1_2_7_2_2"/>
<font color="Maroon" title="c17">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">ralpha</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c19">ralpha</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">y1</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c19">ralpha</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c19">ralpha</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="xcmplx_1.html#T88" title="XCMPLX_1:th.88">XCMPLX_1:88</a>, <a class="txt" href="mesfunc5.html#E7:105_1_2_7_2_2"><i><font color="Green" title="E131">A319</font></i></a></i>;<br/>
<a NAME="E11:105_1_2_7_2_2"/><b>then </b>
<font color="Maroon" title="c17">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <span class="p2">(<span class="default"><font color="Maroon" title="c19">ralpha</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c19">ralpha</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span></span>)</span></span>)</span> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <span class="p2">(<span class="default"><font color="Maroon" title="c21">y1</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p3">(<span class="default"><font color="Maroon" title="c19">ralpha</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c19">ralpha</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="extreal1.html#T38" title="EXTREAL1:th.38">EXTREAL1:38</a></i>;<br/>
<a NAME="E12:105_1_2_7_2_2"/><b>then </b><i><font color="Green" title="E133">A321</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <span class="p3">(<span class="default"><a href="seq_4.html#K3" title="SEQ_4:func.3">min</a> <font color="Maroon" title="c10">RINGA</font></span>)</span></span>)</span> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <span class="p4">(<span class="default"><a href="seq_4.html#K3" title="SEQ_4:func.3">min</a> <font color="Maroon" title="c10">RINGA</font></span>)</span></span>)</span> <a href="extreal1.html#K3" title="EXTREAL1:func.3">/</a> <span class="p3">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> 2</span>)</span></span>)</span></span>)</span> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c17">y</font></span>)</span> <a href="extreal1.html#K3" title="EXTREAL1:func.3">/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <span class="p4">(<span class="default"><a href="seq_4.html#K3" title="SEQ_4:func.3">min</a> <font color="Maroon" title="c10">RINGA</font></span>)</span></span>)</span> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <span class="p5">(<span class="default"><a href="seq_4.html#K3" title="SEQ_4:func.3">min</a> <font color="Maroon" title="c10">RINGA</font></span>)</span></span>)</span> <a href="extreal1.html#K3" title="EXTREAL1:func.3">/</a> <span class="p4">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> 2</span>)</span></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">y</font>
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E6:105_1_2_7_2_2"><i><font color="Green" title="E130">A318</font></i></a>, <a class="txt" href="mesfunc5.html#E7:105_1_2_7_2_2"><i><font color="Green" title="E131">A319</font></i></a>, <a class="ref" href="extreal1.html#T39" title="EXTREAL1:th.39">EXTREAL1:39</a></i>;<br/>
<a NAME="E13:105_1_2_7_2_2"/><i><font color="Green" title="E134">A322</font></i>: 
<font color="Maroon" title="c17">y</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">ralpha</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c19">ralpha</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span></span>)</span> is   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>
 
<b>by </b><i><a class="ref" href="xreal_0.html#D1" title="XREAL_0:def.1">XREAL_0:def 1</a></i>;<br/>
<font color="Maroon" title="c17">y</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">ralpha</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c19">ralpha</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c21">y1</font></span>)</span> <a href="extreal1.html#K3" title="EXTREAL1:func.3">/</a> <span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <span class="p2">(<span class="default"><font color="Maroon" title="c19">ralpha</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c19">ralpha</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="mesfunc5.html#E7:105_1_2_7_2_2"><i><font color="Green" title="E131">A319</font></i></a>, <a class="ref" href="extreal1.html#T39" title="EXTREAL1:th.39">EXTREAL1:39</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c17">y</font></span>)</span> <a href="extreal1.html#K3" title="EXTREAL1:func.3">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <span class="p3">(<span class="default"><a href="seq_4.html#K3" title="SEQ_4:func.3">min</a> <font color="Maroon" title="c10">RINGA</font></span>)</span></span>)</span> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <span class="p4">(<span class="default"><a href="seq_4.html#K3" title="SEQ_4:func.3">min</a> <font color="Maroon" title="c10">RINGA</font></span>)</span></span>)</span> <a href="extreal1.html#K3" title="EXTREAL1:func.3">/</a> <span class="p3">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> 2</span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="mesfunc5.html#E3:105_1_2_7_2_2"><i><font color="Green" title="E127">A315</font></i></a>, <a class="ref" href="supinf_2.html#T5" title="SUPINF_2:th.5">SUPINF_2:5</a></i>
;<br/>
<a NAME="E15:105_1_2_7_2_2"/><b>then </b><i><font color="Green" title="E135">A323</font></i>: 
<span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c17">y</font></span>)</span> <a href="extreal1.html#K3" title="EXTREAL1:func.3">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <span class="p3">(<span class="default"><a href="seq_4.html#K3" title="SEQ_4:func.3">min</a> <font color="Maroon" title="c10">RINGA</font></span>)</span></span>)</span> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <span class="p4">(<span class="default"><a href="seq_4.html#K3" title="SEQ_4:func.3">min</a> <font color="Maroon" title="c10">RINGA</font></span>)</span></span>)</span> <a href="extreal1.html#K3" title="EXTREAL1:func.3">/</a> <span class="p3">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> 2</span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a> 
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E13:105_1_2_7_2_2"><i><font color="Green" title="E134">A322</font></i></a>, <a class="ref" href="xxreal_0.html#T9" title="XXREAL_0:th.9">XXREAL_0:9</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c22">H</font> being   <a href="prob_1.html#NM2" title="PROB_1:NM.2">SetSequence</a> of <font color="Maroon" title="c1">X</font>, <font color="Maroon" title="c23">MF</font> being   <a href="mesfunc5.html#NM1" title="MESFUNC5:NM.1">ExtREAL_sequence</a><b> such that </b><br/><a NAME="E17:105_1_2_7_2_2"/><i><font color="Green" title="E136">A324</font></i>: 
for <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <font color="Maroon" title="c22">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mesfunc1.html#K15" title="MESFUNC1:func.15">less_dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="mesfunc1.html#K6" title="MESFUNC1:func.6">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Olive" title="b1">n</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <span class="p3">(<span class="default"><a href="seq_4.html#K3" title="SEQ_4:func.3">min</a> <font color="Maroon" title="c10">RINGA</font></span>)</span></span>)</span> <a href="extreal1.html#K3" title="EXTREAL1:func.3">/</a> <span class="p2">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> 2</span>)</span></span>)</span>
 <b>and </b><br/><a NAME="E18:105_1_2_7_2_2"/><i><font color="Green" title="E137">A325</font></i>: 
for <font color="Olive" title="b1">n</font>, <font color="Olive" title="b2">m</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">m</font> holds <br/><font color="Maroon" title="c22">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c22">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">m</font>
 <b>and </b><br/><a NAME="E19:105_1_2_7_2_2"/>
for <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <font color="Maroon" title="c22">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">g</font>
 <b>and </b><br/><a NAME="E20:105_1_2_7_2_2"/><i><font color="Green" title="E138">A326</font></i>: 
for <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <font color="Maroon" title="c23">MF</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">M</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c22">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font></span>)</span>
 <b>and </b><br/><a NAME="E21:105_1_2_7_2_2"/><i><font color="Green" title="E139">A327</font></i>: 
<font color="Maroon" title="c3">M</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">g</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_2.html#K8" title="SUPINF_2:func.8">sup</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K19" title="SUPINF_2:func.19">rng</a> <font color="Maroon" title="c23">MF</font></span>)</span>
 <b>and </b><br/><a NAME="E22:105_1_2_7_2_2"/><i><font color="Green" title="E140">A328</font></i>: 
for <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <font color="Maroon" title="c22">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">S</font>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E4:105_1_2_7_2_2"><i><font color="Green" title="E128">A316</font></i></a>, <a class="txt" href="mesfunc5.html#E5:105_1_2_7_2_2"><i><font color="Green" title="E129">A317</font></i></a>, <a class="txt" href="mesfunc5.html#E31:105_1_2"><i><font color="Green" title="E121">A37</font></i></a></i>;<br/>
<a NAME="E23:105_1_2_7_2_2"/>
 ex <font color="Olive" title="b1">z</font> being <a href="xxreal_0.html#V1" title="XXREAL_0:attr.1">ext-real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>  st <br/>( <font color="Olive" title="b1">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="supinf_2.html#K19" title="SUPINF_2:func.19">rng</a> <font color="Maroon" title="c23">MF</font> &amp; <span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c17">y</font></span>)</span> <a href="extreal1.html#K3" title="EXTREAL1:func.3">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <span class="p3">(<span class="default"><a href="seq_4.html#K3" title="SEQ_4:func.3">min</a> <font color="Maroon" title="c10">RINGA</font></span>)</span></span>)</span> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <span class="p4">(<span class="default"><a href="seq_4.html#K3" title="SEQ_4:func.3">min</a> <font color="Maroon" title="c10">RINGA</font></span>)</span></span>)</span> <a href="extreal1.html#K3" title="EXTREAL1:func.3">/</a> <span class="p3">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> 2</span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">z</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="105_1_2_7_2_2_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:105_1_2_7_2_2_2"/>
for <font color="Olive" title="b1">z</font> being <a href="xxreal_0.html#V1" title="XXREAL_0:attr.1">ext-real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>   holds <br/>(  not <font color="Olive" title="b1">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="supinf_2.html#K19" title="SUPINF_2:func.19">rng</a> <font color="Maroon" title="c23">MF</font> or  not <span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c17">y</font></span>)</span> <a href="extreal1.html#K3" title="EXTREAL1:func.3">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <span class="p3">(<span class="default"><a href="seq_4.html#K3" title="SEQ_4:func.3">min</a> <font color="Maroon" title="c10">RINGA</font></span>)</span></span>)</span> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <span class="p4">(<span class="default"><a href="seq_4.html#K3" title="SEQ_4:func.3">min</a> <font color="Maroon" title="c10">RINGA</font></span>)</span></span>)</span> <a href="extreal1.html#K3" title="EXTREAL1:func.3">/</a> <span class="p3">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> 2</span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">z</font> )
 ;<br/>

<a NAME="E2:105_1_2_7_2_2_2"/><b>then </b>
for <font color="Olive" title="b1">z</font> being <a href="xxreal_0.html#V1" title="XXREAL_0:attr.1">ext-real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>   st <font color="Olive" title="b1">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="supinf_2.html#K19" title="SUPINF_2:func.19">rng</a> <font color="Maroon" title="c23">MF</font> holds <br/><font color="Olive" title="b1">z</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c17">y</font></span>)</span> <a href="extreal1.html#K3" title="EXTREAL1:func.3">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <span class="p3">(<span class="default"><a href="seq_4.html#K3" title="SEQ_4:func.3">min</a> <font color="Maroon" title="c10">RINGA</font></span>)</span></span>)</span> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <span class="p4">(<span class="default"><a href="seq_4.html#K3" title="SEQ_4:func.3">min</a> <font color="Maroon" title="c10">RINGA</font></span>)</span></span>)</span> <a href="extreal1.html#K3" title="EXTREAL1:func.3">/</a> <span class="p3">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> 2</span>)</span></span>)</span></span>)</span>
 
;<br/>
<a NAME="E3:105_1_2_7_2_2_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c17">y</font></span>)</span> <a href="extreal1.html#K3" title="EXTREAL1:func.3">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <span class="p3">(<span class="default"><a href="seq_4.html#K3" title="SEQ_4:func.3">min</a> <font color="Maroon" title="c10">RINGA</font></span>)</span></span>)</span> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <span class="p4">(<span class="default"><a href="seq_4.html#K3" title="SEQ_4:func.3">min</a> <font color="Maroon" title="c10">RINGA</font></span>)</span></span>)</span> <a href="extreal1.html#K3" title="EXTREAL1:func.3">/</a> <span class="p3">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> 2</span>)</span></span>)</span></span>)</span> is    <a href="supinf_1.html#M1" title="SUPINF_1:mode.1">majorant</a> of  <a href="supinf_2.html#K19" title="SUPINF_2:func.19">rng</a> <font color="Maroon" title="c23">MF</font>
 
<b>by </b><i><a class="ref" href="supinf_1.html#D9" title="SUPINF_1:def.9">SUPINF_1:def 9</a></i>;<br/>
<b>hence </b><a NAME="E4:105_1_2_7_2_2_2"/>
contradiction
 <b>by </b><i><a class="txt" href="mesfunc5.html#E15:105_1_2_7_2_2"><i><font color="Green" title="E135">A323</font></i></a>, <a class="txt" href="mesfunc5.html#E21:105_1_2_7_2_2"><i><font color="Green" title="E139">A327</font></i></a>, <a class="txt" href="mesfunc5.html#E1:105_1_2_7_2"><i><font color="Green" title="E125">A280</font></i></a>, <a class="ref" href="supinf_1.html#D16" title="SUPINF_1:def.16">SUPINF_1:def 16</a></i>;<br/>


</div><b>end;</b></div>
<b>then consider </b><font color="Maroon" title="c24">z</font> being   <a href="supinf_1.html#NM3" title="SUPINF_1:NM.3">R_eal</a><b> such that </b><br/><a NAME="E25:105_1_2_7_2_2"/><i><font color="Green" title="E141">A329</font></i>: 
( <font color="Maroon" title="c24">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="supinf_2.html#K19" title="SUPINF_2:func.19">rng</a> <font color="Maroon" title="c23">MF</font> &amp; <span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c17">y</font></span>)</span> <a href="extreal1.html#K3" title="EXTREAL1:func.3">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <span class="p3">(<span class="default"><a href="seq_4.html#K3" title="SEQ_4:func.3">min</a> <font color="Maroon" title="c10">RINGA</font></span>)</span></span>)</span> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <span class="p4">(<span class="default"><a href="seq_4.html#K3" title="SEQ_4:func.3">min</a> <font color="Maroon" title="c10">RINGA</font></span>)</span></span>)</span> <a href="extreal1.html#K3" title="EXTREAL1:func.3">/</a> <span class="p3">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> 2</span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c24">z</font> )
 ;<br/>
<b>consider </b><font color="Maroon" title="c25">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E27:105_1_2_7_2_2"/><i><font color="Green" title="E142">A330</font></i>: 
( <font color="Maroon" title="c25">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c23">MF</font> &amp; <font color="Maroon" title="c24">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">MF</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c25">x</font> )
 <b>by </b><i><a class="txt" href="mesfunc5.html#E25:105_1_2_7_2_2"><i><font color="Green" title="E141">A329</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c26">N0</font> = <font color="Maroon" title="c25">x</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="txt" href="mesfunc5.html#E27:105_1_2_7_2_2"><i><font color="Green" title="E142">A330</font></i></a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c26">N0</font>
;<br/>

<b>thus </b><a NAME="E29:105_1_2_7_2_2"/>
for <font color="Olive" title="b1">m</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Maroon" title="c26">N0</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">m</font> holds <br/><font color="Maroon" title="c17">y</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">L</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Olive" title="b1">m</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="105_1_2_7_2_2_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c27">m</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:105_1_2_7_2_2_3"/><i><font color="Green" title="E143">A331</font></i>: 
<font color="Maroon" title="c26">N0</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c27">m</font>
 ;<br/>

<a NAME="E2:105_1_2_7_2_2_3"/>
<span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c17">y</font></span>)</span> <a href="extreal1.html#K3" title="EXTREAL1:func.3">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <span class="p3">(<span class="default"><a href="seq_4.html#K3" title="SEQ_4:func.3">min</a> <font color="Maroon" title="c10">RINGA</font></span>)</span></span>)</span> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <span class="p4">(<span class="default"><a href="seq_4.html#K3" title="SEQ_4:func.3">min</a> <font color="Maroon" title="c10">RINGA</font></span>)</span></span>)</span> <a href="extreal1.html#K3" title="EXTREAL1:func.3">/</a> <span class="p3">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> 2</span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">M</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c22">H</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Maroon" title="c26">N0</font></span>)</span>
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E20:105_1_2_7_2_2"><i><font color="Green" title="E138">A326</font></i></a>, <a class="txt" href="mesfunc5.html#E27:105_1_2_7_2_2"><i><font color="Green" title="E142">A330</font></i></a>, <a class="txt" href="mesfunc5.html#E25:105_1_2_7_2_2"><i><font color="Green" title="E141">A329</font></i></a></i>;<br/>
<a NAME="E3:105_1_2_7_2_2_3"/><b>then </b><i><font color="Green" title="E144">A332</font></i>: 
 <a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c17">y</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <span class="p3">(<span class="default"><a href="seq_4.html#K3" title="SEQ_4:func.3">min</a> <font color="Maroon" title="c10">RINGA</font></span>)</span></span>)</span> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <span class="p4">(<span class="default"><a href="seq_4.html#K3" title="SEQ_4:func.3">min</a> <font color="Maroon" title="c10">RINGA</font></span>)</span></span>)</span> <a href="extreal1.html#K3" title="EXTREAL1:func.3">/</a> <span class="p3">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> 2</span>)</span></span>)</span></span>)</span> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c22">H</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Maroon" title="c26">N0</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E8:105_1_2_7_2_2"><i><font color="Green" title="E132">A320</font></i></a>, <a class="txt" href="mesfunc5.html#E12:105_1_2_7_2_2"><i><font color="Green" title="E133">A321</font></i></a>, <a class="ref" href="extreal1.html#T42" title="EXTREAL1:th.42">EXTREAL1:42</a></i>;<br/>
<a NAME="E4:105_1_2_7_2_2_3"/><i><font color="Green" title="E145">A333</font></i>: 
<font color="Maroon" title="c22">H</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Maroon" title="c26">N0</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c22">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c27">m</font>
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E18:105_1_2_7_2_2"><i><font color="Green" title="E137">A325</font></i></a>, <a class="txt" href="mesfunc5.html#E1:105_1_2_7_2_2_3"><i><font color="Green" title="E143">A331</font></i></a></i>;<br/>
<a NAME="E5:105_1_2_7_2_2_3"/>
( <font color="Maroon" title="c22">H</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Maroon" title="c26">N0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">S</font> &amp; <font color="Maroon" title="c22">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c27">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">S</font> )
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E22:105_1_2_7_2_2"><i><font color="Green" title="E140">A328</font></i></a></i>;<br/>
<a NAME="E6:105_1_2_7_2_2_3"/><b>then </b>
<font color="Maroon" title="c3">M</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c22">H</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Maroon" title="c26">N0</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">M</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c22">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c27">m</font></span>)</span>
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E4:105_1_2_7_2_2_3"><i><font color="Green" title="E145">A333</font></i></a>, <a class="ref" href="measure1.html#T62" title="MEASURE1:th.62">MEASURE1:62</a></i>;<br/>
<a NAME="E7:105_1_2_7_2_2_3"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <span class="p3">(<span class="default"><a href="seq_4.html#K3" title="SEQ_4:func.3">min</a> <font color="Maroon" title="c10">RINGA</font></span>)</span></span>)</span> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <span class="p4">(<span class="default"><a href="seq_4.html#K3" title="SEQ_4:func.3">min</a> <font color="Maroon" title="c10">RINGA</font></span>)</span></span>)</span> <a href="extreal1.html#K3" title="EXTREAL1:func.3">/</a> <span class="p3">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> 2</span>)</span></span>)</span></span>)</span> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c22">H</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Maroon" title="c26">N0</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <span class="p3">(<span class="default"><a href="seq_4.html#K3" title="SEQ_4:func.3">min</a> <font color="Maroon" title="c10">RINGA</font></span>)</span></span>)</span> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <span class="p4">(<span class="default"><a href="seq_4.html#K3" title="SEQ_4:func.3">min</a> <font color="Maroon" title="c10">RINGA</font></span>)</span></span>)</span> <a href="extreal1.html#K3" title="EXTREAL1:func.3">/</a> <span class="p3">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> 2</span>)</span></span>)</span></span>)</span> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c22">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c27">m</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E8:105_1_2_7_2_2"><i><font color="Green" title="E132">A320</font></i></a>, <a class="ref" href="extreal1.html#T42" title="EXTREAL1:th.42">EXTREAL1:42</a></i>;<br/>
<a NAME="E8:105_1_2_7_2_2_3"/><b>then </b>
 <a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c17">y</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <span class="p3">(<span class="default"><a href="seq_4.html#K3" title="SEQ_4:func.3">min</a> <font color="Maroon" title="c10">RINGA</font></span>)</span></span>)</span> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <span class="p4">(<span class="default"><a href="seq_4.html#K3" title="SEQ_4:func.3">min</a> <font color="Maroon" title="c10">RINGA</font></span>)</span></span>)</span> <a href="extreal1.html#K3" title="EXTREAL1:func.3">/</a> <span class="p3">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> 2</span>)</span></span>)</span></span>)</span> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c22">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c27">m</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E3:105_1_2_7_2_2_3"><i><font color="Green" title="E144">A332</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E9:105_1_2_7_2_2_3"/><b>then </b><i><font color="Green" title="E146">A334</font></i>: 
 <a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c17">y</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <span class="p3">(<span class="default"><a href="seq_4.html#K3" title="SEQ_4:func.3">min</a> <font color="Maroon" title="c10">RINGA</font></span>)</span></span>)</span> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <span class="p4">(<span class="default"><a href="seq_4.html#K3" title="SEQ_4:func.3">min</a> <font color="Maroon" title="c10">RINGA</font></span>)</span></span>)</span> <a href="extreal1.html#K3" title="EXTREAL1:func.3">/</a> <span class="p3">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> 2</span>)</span></span>)</span></span>)</span> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <span class="p2">(<span class="default"><a href="mesfunc1.html#K15" title="MESFUNC1:func.15">less_dom</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="mesfunc1.html#K6" title="MESFUNC1:func.6">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c27">m</font></span>)</span></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <span class="p5">(<span class="default"><a href="seq_4.html#K3" title="SEQ_4:func.3">min</a> <font color="Maroon" title="c10">RINGA</font></span>)</span></span>)</span> <a href="extreal1.html#K3" title="EXTREAL1:func.3">/</a> <span class="p4">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> 2</span>)</span></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E17:105_1_2_7_2_2"><i><font color="Green" title="E136">A324</font></i></a></i>;<br/>
<a NAME="E10:105_1_2_7_2_2_3"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <span class="p3">(<span class="default"><a href="seq_4.html#K3" title="SEQ_4:func.3">min</a> <font color="Maroon" title="c10">RINGA</font></span>)</span></span>)</span> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <span class="p4">(<span class="default"><a href="seq_4.html#K3" title="SEQ_4:func.3">min</a> <font color="Maroon" title="c10">RINGA</font></span>)</span></span>)</span> <a href="extreal1.html#K3" title="EXTREAL1:func.3">/</a> <span class="p3">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> 2</span>)</span></span>)</span></span>)</span> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <span class="p2">(<span class="default"><a href="mesfunc1.html#K15" title="MESFUNC1:func.15">less_dom</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="mesfunc1.html#K6" title="MESFUNC1:func.6">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c27">m</font></span>)</span></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <span class="p5">(<span class="default"><a href="seq_4.html#K3" title="SEQ_4:func.3">min</a> <font color="Maroon" title="c10">RINGA</font></span>)</span></span>)</span> <a href="extreal1.html#K3" title="EXTREAL1:func.3">/</a> <span class="p4">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> 2</span>)</span></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="mesfunc5.html#K5" title="MESFUNC5:func.5">integral'</a> <font color="Maroon" title="c3">M</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c27">m</font></span>)</span>
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E3:105_1_2_7_2"><i><font color="Green" title="E126">A281</font></i></a>, <a class="txt" href="mesfunc5.html#E4:105_1_2_7_2_2"><i><font color="Green" title="E128">A316</font></i></a>, <a class="txt" href="mesfunc5.html#E5:105_1_2_7_2_2"><i><font color="Green" title="E129">A317</font></i></a></i>;<br/>
<a NAME="E11:105_1_2_7_2_2_3"/><b>then </b>
 <a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c17">y</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="mesfunc5.html#K5" title="MESFUNC5:func.5">integral'</a> <font color="Maroon" title="c3">M</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c27">m</font></span>)</span>
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a>, <a class="txt" href="mesfunc5.html#E9:105_1_2_7_2_2_3"><i><font color="Green" title="E146">A334</font></i></a></i>;<br/>
<b>hence </b><a NAME="E12:105_1_2_7_2_2_3"/>
<font color="Maroon" title="c17">y</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">L</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c27">m</font>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E8:105"><i><font color="Green" title="E106">A8</font></i></a></i>;<br/>


</div><b>end;</b></div>


</div><b>end;</b></div><a NAME="E5:105_1_2_7_2"/><b>then </b><i><font color="Green" title="E127">A335</font></i>: 
<font color="Maroon" title="c6">L</font> is <a href="mesfunc5.html#V8" title="MESFUNC5:attr.8">convergent_to_+infty</a>
 <b>by </b><i><a class="ref" href="mesfunc5.html#D9" target="_self" title="MESFUNC5:def.9">Def9</a></i>;<br/><b>hence </b><a NAME="E6:105_1_2_7_2"/>
<font color="Maroon" title="c6">L</font> is <a href="mesfunc5.html#V10" title="MESFUNC5:attr.10">convergent</a>
 <b>by </b><i><a class="ref" href="mesfunc5.html#D11" target="_self" title="MESFUNC5:def.11">Def11</a></i>;<br/><a NAME="E7:105_1_2_7_2"/><b>then </b>
(  ex <font color="Olive" title="b1">g</font> being <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>  st <br/>(  <a href="mesfunc5.html#K2" title="MESFUNC5:func.2">lim</a> <font color="Maroon" title="c6">L</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">g</font> &amp; ( for <font color="Olive" title="b2">p</font> being <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>   st 0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Olive" title="b2">p</font> holds <br/> ex <font color="Olive" title="b3">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <br/>for <font color="Olive" title="b4">m</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b3">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b4">m</font> holds <br/><span class="p1"><a href="extreal1.html#K4" title="EXTREAL1:func.4">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">L</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Olive" title="b4">m</font></span>)</span> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <span class="p3">(<span class="default"><a href="mesfunc5.html#K2" title="MESFUNC5:func.2">lim</a> <font color="Maroon" title="c6">L</font></span>)</span></span>)</span></span><a href="extreal1.html#K4" title="EXTREAL1:func.4">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b2">p</font> ) &amp; <font color="Maroon" title="c6">L</font> is <a href="mesfunc5.html#V7" title="MESFUNC5:attr.7">convergent_to_finite_number</a> ) or (  <a href="mesfunc5.html#K2" title="MESFUNC5:func.2">lim</a> <font color="Maroon" title="c6">L</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>  &amp; <font color="Maroon" title="c6">L</font> is <a href="mesfunc5.html#V8" title="MESFUNC5:attr.8">convergent_to_+infty</a> ) or (  <a href="mesfunc5.html#K2" title="MESFUNC5:func.2">lim</a> <font color="Maroon" title="c6">L</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a>  &amp; <font color="Maroon" title="c6">L</font> is <a href="mesfunc5.html#V9" title="MESFUNC5:attr.9">convergent_to_-infty</a> ) )
 <b>by </b><i><a class="ref" href="mesfunc5.html#D12" target="_self" title="MESFUNC5:def.12">Def12</a></i>;<br/><b>hence </b><a NAME="E8:105_1_2_7_2"/>
 <a href="mesfunc5.html#K5" title="MESFUNC5:func.5">integral'</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">g</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="mesfunc5.html#K2" title="MESFUNC5:func.2">lim</a> <font color="Maroon" title="c6">L</font>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E5:105_1_2_7_2"><i><font color="Green" title="E127">A335</font></i></a>, <a class="ref" href="mesfunc5.html#T56" target="_self" title="MESFUNC5:th.56">Th56</a>, <a class="ref" href="xxreal_0.html#T4" title="XXREAL_0:th.4">XXREAL_0:4</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>

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